Prerequisites

# Radicals and Rational Exponents

### Learning Objectives

In this section students will:

- Evaluate square roots.
- Use the product rule to simplify square roots.
- Use the quotient rule to simplify square roots.
- Add and subtract square roots.
- Rationalize denominators.
- Use rational roots.

A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in (Figure), and use the Pythagorean Theorem.

### Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Sincethe square root ofisThe square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.

In general terms, ifis a positive real number, then the square root ofis a number that, when multiplied by itself, givesThe square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equalsThe square root obtained using a calculator is the principal square root.

The principal square root ofis written asThe symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.

### Principal Square Root

The principal square root ofis the nonnegative number that, when multiplied by itself, equalsIt is written as a radical expression**,** with a symbol called a radical over the term called the radicand:

**Does**

*No. Although both**and**are**the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is*

### Evaluating Square Roots

Evaluate each expression.

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- because
- becauseand
- because
- becauseand

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**Forcan we find the square roots before adding?**

*No.**This is not equivalent to**The order of operations requires us to add the terms in the radicand before finding the square root.*

### Try It

Evaluate each expression.

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### Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the *product rule for simplifying square roots,* which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewriteasWe can also use the product rule to express the product of multiple radical expressions as a single radical expression.

### The Product Rule for Simplifying Square Roots

Ifandare nonnegative, the square root of the productis equal to the product of the square roots ofand

### How To

**Given a square root radical expression, use the product rule to simplify it.
**

- Factor any perfect squares from the radicand.
- Write the radical expression as a product of radical expressions.
- Simplify.

### Using the Product Rule to Simplify Square Roots

Simplify the radical expression.

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### Try It

Simplify

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Notice the absolute value signs around *x* and *y*? Thatâ€™s because their value must be positive!

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### How To

**Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.
**

- Express the product of multiple radical expressions as a single radical expression.
- Simplify.

### Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression.

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### Try It

Simplifyassuming

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### Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the *quotient rule for simplifying square roots.* It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewriteas

### The Quotient Rule for Simplifying Square Roots

The square root of the quotientis equal to the quotient of the square roots ofandwhere

### How To

**Given a radical expression, use the quotient rule to simplify it.
**

- Write the radical expression as the quotient of two radical expressions.
- Simplify the numerator and denominator.

### Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

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### Try It

Simplify

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We do not need the absolute value signs forbecause that term will always be nonnegative.

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### Using the Quotient Rule to Simplify an Expression with Two Square Roots

Simplify the radical expression.

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### Try It

Simplify

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### Adding and Subtracting Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum ofandisHowever, it is often possible to simplify radical expressions, and that may change the radicand. The radical expressioncan be written with ain the radicand, asso

### How To

**Given a radical expression requiring addition or subtraction of square roots, solve.**

- Simplify each radical expression.
- Add or subtract expressions with equal radicands.

### Adding Square Roots

Add

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We can rewriteasAccording the product rule, this becomesThe square root ofis 2, so the expression becomeswhich isNow we can the terms have the same radicand so we can add.

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### Try It

Add

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### Subtracting Square Roots

Subtract

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Rewrite each term so they have equal radicands.

Now the terms have the same radicand so we can subtract.

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### Try It

Subtract

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### Rationalizing Denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called *rationalizing the denominator*.

We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.

For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator ismultiply by

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator isthen the conjugate is

### How To

**Given an expression with a single square root radical term in the denominator, rationalize the denominator.**

- Multiply the numerator and denominator by the radical in the denominator.
- Simplify.

### Rationalizing a Denominator Containing a Single Term

Writein simplest form.

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The radical in the denominator isSo multiply the fraction byThen simplify.

### Try It

Writein simplest form.

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### How To

**Given an expression with a radical term and a constant in the denominator, rationalize the denominator.**

- Find the conjugate of the denominator.
- Multiply the numerator and denominator by the conjugate.
- Use the distributive property.
- Simplify.

### Rationalizing a Denominator Containing Two Terms

Writein simplest form.

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Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate ofisThen multiply the fraction by

### Try It

Writein simplest form.

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### Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

#### Understanding *n*th Roots

Suppose we know thatWe want to find what number raised to the 3rd power is equal to 8. Sincewe say that 2 is the cube root of 8.

The *n*th root ofis a number that, when raised to the *n*th power, givesFor example,is the 5th root ofbecauseIfis a real number with at least one *n*th root, then the principal *n*th root ofis the number with the same sign asthat, when raised to the *n*th power, equals

The principal *n*th root ofis written aswhereis a positive integer greater than or equal to 2. In the radical expression,is called the index of the radical.

### Principal *n*th Root

Ifis a real number with at least one *n*th root, then the principal *n*th root ofwritten asis the number with the same sign asthat, when raised to the *n*th power, equalsThe index of the radical is

### Simplifying *n*th Roots

Simplify each of the following:

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- because
- First, express the product as a single radical expression.because

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### Try It

Simplify.

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#### Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the indexis even, thencannot be negative.

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an *n*th root. The numerator tells us the power and the denominator tells us the root.

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

### Rational Exponents

Rational exponents are another way to express principal *n*th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

### How To

**Given an expression with a rational exponent, write the expression as a radical.**

- Determine the power by looking at the numerator of the exponent.
- Determine the root by looking at the denominator of the exponent.
- Using the base as the radicand, raise the radicand to the power and use the root as the index.

### Writing Rational Exponents as Radicals

Writeas a radical. Simplify.

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The 2 tells us the power and the 3 tells us the root.

We know thatbecauseBecause the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

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### Try It

Writeas a radical. Simplify.

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### Writing Radicals as Rational Exponents

Writeusing a rational exponent.

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### Try It

Writeusing a rational exponent.

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### Simplifying Rational Exponents

Simplify:

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### Try It

Simplify

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Access these online resources for additional instruction and practice with radicals and rational exponents.

### Key Concepts

- The principal square root of a numberis the nonnegative number that when multiplied by itself equalsSee (Figure).
- Ifandare nonnegative, the square root of the productis equal to the product of the square roots ofandSee (Figure) and (Figure).
- Ifandare nonnegative, the square root of the quotientis equal to the quotient of the square roots ofandSee (Figure) and (Figure).
- We can add and subtract radical expressions if they have the same radicand and the same index. See (Figure) and (Figure).
- Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See (Figure) and (Figure).
- The principal
*n*th root ofis the number with the same sign asthat when raised to the*n*th power equalsThese roots have the same properties as square roots. See (Figure). - Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See (Figure) and (Figure).
- The properties of exponents apply to rational exponents. See (Figure).

### Section Exercises

#### Verbal

What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

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When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

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Where would radicals come in the order of operations? Explain why.

Every number will have two square roots. What is the principal square root?

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The principal square root is the nonnegative root of the number.

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Can a radical with a negative radicand have a real square root? Why or why not?

#### Numeric

For the following exercises, simplify each expression.

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16

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10

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14

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25

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#### Algebraic

For the following exercises, simplify each expression.

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#### Real-World Applications

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluatingWhat is the length of the guy wire?

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500 feet

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A car accelerates at a rate ofwhere *t* is the time in seconds after the car moves from rest. Simplify the expression.

#### Extensions

For the following exercises, simplify each expression.

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### Glossary

- index
- the number above the radical sign indicating the
*n*th root

- principal
*n*th root - the number with the same sign asthat when raised to the
*n*th power equals

- principal square root
- the nonnegative square root of a numberthat, when multiplied by itself, equals

- radical
- the symbol used to indicate a root

- radical expression
- an expression containing a radical symbol

- radicand
- the number under the radical symbol