Decoding Rational Exponents and Radicals: A Comprehensive Guide

Power	Multiplication	Value
$4^{2}$	$4 \cdot 4$	$16$
$2^{3}$	$2 \cdot 2 \cdot 2$	$8$
$5^{4}$	$5 \cdot 5 \cdot 5 \cdot 5$	$625$
$1^{5}$	$1 \cdot 1 \cdot 1 \cdot 1 \cdot 1$	$1$

Power

Multiplication

Value

4^{2}

4 \cdot 4

16

2^{3}

2 \cdot 2 \cdot 2

8

5^{4}

5 \cdot 5 \cdot 5 \cdot 5

625

1^{5}

1 \cdot 1 \cdot 1 \cdot 1 \cdot 1

1

Rewrite the power as a radical.

2^{\frac{1}{2}}

5^{\frac{1}{3}}

6^{\frac{2}{7}}

Examining the power, we can see that the denominator of the exponent is 2. Any power with a rational exponent where the denominator is 2 can be written as the square root of the base raised to the numerator of the exponent.

Note that the numerator of our exponent is 1, and since any value raised to the power of 1 equals that value, the answer is sqrt(2).

In this case, the denominator of the exponent is 3. This means that we can rewrite this expression as the cube root of 5.

The rational exponent has a numerator that is not 1. This indicates that we will obtain two acceptable equivalent answers.

The first acceptable answer is sqrt(6^2). If we calculate the power, we can obtain a second equivalent expression.

Finally, when rewriting a rational exponent as a root, we can take the seventh root of 6 and then raise to the power of 2. 6^(27) = (sqrt(6))^2 Therefore, sqrt(6^2), sqrt(36), and (sqrt(6))^2 are all acceptable answers.

Rewrite the radical as a power with a rational exponent.

3

46

5 4^{3}

5 2^{a}

Examining the radical, we notice that it does not show an index. This actually implies that the index is 2. sqrt(3) ⇔ sqrt(3) Therefore, we can write the radical expression as a power by raising 3 to the power of 12.

In this case, the index of the radical is 4. sqrt(6) To rewrite the expression as a power with a rational exponent, we must raise 6 to the power of 14.

The radical has an index of 5. Also, the radicand is a power with exponent 3. We can rewrite the expression by recalling that the index of a radical expression is the denominator of the rational exponent. Furthermore, the exponent of the radicand is the numerator of the rational exponent.

However, we can obtain a different but equivalent form by first calculating the value of 4^3.

These expressions are equivalent, meaning that both answers are correct. 4^(35) ⇔ 64^(15)

Just like in the previous part, the index of a radical expression is the denominator of the rational exponent. Also, the exponent of the radicand is the numerator of the rational exponent. Let's use this information to rewrite the expression.

Simplify the following expressions.

(3^{\frac{1}{2}})^{2}

(45)^{4}

(34)^{9}

(2^{12})^{\frac{1}{4}}

Let's start by noting that raising a number to the power of 12 is equivalent to calculating the square root of the number.

As such, taking the square root of the number and raising to the power of 2 are inverse operations. This means that they undo each other. Therefore, if we calculate the square root of a number and then raise the result to the power of 2, we end up with the same number.

Calculating the fourth root of a number and raising the number to the power of 4 are inverse operations, meaning that they undo each other. Therefore, if we calculate the fourth root of a number and then raise the result to the power of 4, we end up with the same number.

This time, we will start by rewriting the expression somewhat so that we can simplify it.

The number 2^(12) is raised to the power of 14. This can be rewritten as the fourth root of the number.

Solve the following equations. Write the solutions as powers with rational exponents in their simplest form. Make sure all solutions are written.

x^{3} = 12

x^{4} = 7

On the left-hand side of the equation, the variable x is cubed . To eliminate this exponent and thereby isolate x, we must perform an operation that cancels out the cube. To do this, we will raise the expression to the power of 13. This is the same thing as taking the cube root of the expression. Since this is an equation, we must perform the operation on both sides to keep both sides equal. Remember to write the final root as a rational exponent.

In this equation, x has been raised to the power of 4. This means that in order to cancel out the exponent on the left-hand side, we must take the fourth root. Note that since the exponent is even, we will obtain two solutions, one positive and one negative.

Select the expression that is not equivalent to the others.

A . B . C . D . (327)^{2} 2 7^{\frac{2}{3}} 3^{2} (227)^{3}

Since some of these numeric expressions have rational exponents and radicals, we will recall one of the definitions of a rational exponent. a^()mn = ( sqrt(a) )^m By using this definition, we can rewrite the expression in B to match the expression in A. 27^()23 = ( sqrt(27) )^2 Furthermore, we can simplify the expression (sqrt(27))^2 by starting with rewriting 27 as a power.

Therefore, the equivalent expressions in A and B are also equivalent to the numeric expression in C. Note that all three expressions are equivalent to 3^2, or 9. Now let's turn our attention to the expression in D. Like we did with the other options, we will start by simplifying the expression as much as possible, then use a calculator to approximate its value. Let's do it!

Now, consider the numeric expression in the parentheses. It can be rewritten as the square root of 2, raised to the power of 3. (2^(32))^3 = ((sqrt(2))^3)^3 Next, the Power of a Power Property for integer exponents can be used. Finally, we can use a calculator.

We can see that since it cannot simplify down to 3^2, or 9, the expression in D is different from the other expressions.

Simplify the following expressions.

2 5^{\frac{1}{2}}

8^{\frac{1}{3}}

(- 243)^{\frac{1}{5}}

From the expression, we can see that 25 has been raised to the power of 12. We can write this rational exponent as a square root.

This time, the denominator of the rational exponent is 3. Raising a number to the power of 13 is equivalent to calculating the cube root of the number.

In this case, the base of our expression is negative. However, the denominator of the rational exponent is odd. This means that when the expression is rewritten as a radical, the index is odd. Therefore, the numeric expression is well defined, since there is no issue in having a root with a negative radicand and an odd index.

If the rational exponent had an even denominator, the expression could not have been calculated. This is because calculating a root with an even index and a negative radicand is not possible — a number multiplied by itself an even number of times will never be negative.

Translating Between Rational Exponents and Radicals

Catch-Up and Review

The Meaning of Exponent

Connection Between Rational Exponents and Radicals

Definition of $a^{\frac{1}{n}}$

$n^{th}$ Root

Rewriting a Power as a Radical

Hint

Solution

Calculating Powers Using Radicals

Finding a Radical and Rewriting It as a Power

Hint

Solution

Definition of a Rational Exponent

Rational Exponent

Another Radical Equivalent to $a^{\frac{m}{n}}$

Expressing a Rational Exponent as a Radical

Hint

Solution

Principal Root

Translating Between Radicals and Rational Exponents

Hint

Solution

Hint

Solution

Translating Between Rational Exponents and Radicals

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