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

Consider the following sphere.

Sphere with a radius of r and a volume of 4pi cm^3

What is the radius

r

of a sphere with volume

4 π

cubic centimeters? Write the answer in exact form.

The volume V of a sphere is calculated by multiplying 43 by π times the cube of the radius. V=4/3π r^3 We are given that the volume of the sphere is 4π. Therefore, by substituting 4π for V, we can apply inverse operations to solve for the radius of the sphere r.

When the radius is sqrt(3) centimeters, the sphere has a volume of 4π cubic centimeters.

Zosia simplified the numeric expression

6 4^{\frac{3}{2}} .

However, according to her teacher, she made a mistake in one of the steps. In what step was a mistake made?

Step 1 Step 2 Step 3 Step 4 6 4^{\frac{3}{2}} (364)^{2} 4^{2} 16

We will look at the steps one at a time.

Step 1

In Step 1, Zosia has copied the numeric expression correctly, so no mistake was made here. Step1 64^(32) ✓

Step 2

Here, Zosia expressed the power with a rational exponent as a root. However, she wrote the numerator of the exponent as the index of the root and the denominator as the exponent. This is her mistake. rcc Step1& 64^(32)& ✓ [0.1em] Step2& (sqrt(64))^2 & * Let's correct Zosia's mistake. Recall that when we rewrite a power with a rational exponent as a root, the numerator and denominator of the rational exponent are the index of the root and the exponent of the expression, respectively. rcc Step1& 64^(32)& ✓ [0.1em] Step2& (sqrt(64))^3 & ✓

Step 3

Here, Zosia calculated the cube root of 64 to be 4. No mistake was made here, but since she made a mistake in the second step, she should have calculated the square root of 64 instead. Let's continue with the correct calculation. Step1& 64^(32) [0.1em] Step2& (sqrt(64))^3 [0.1em] Step3& 8^3

Step 4

What happens in this last step is similar to what happens in the third step. No mistake is made in the calculation itself, but because of the mistake made previously, the number to be calculated is incorrect. The final calculation should be 8^3, not 4^2. Step1& 64^(32) [0.1em] Step2& (sqrt(64))^3 [0.1em] Step3& 8^3 [0.1em] Step4& 512 The final simplification of 64^(32) is 512, not 16.

Using a rational exponent, write an expression for the side length of the square.

The area of a square is calculated by squaring the side length. If the side length of the square is s, we can write the following equation. A=s^2 The area of this square is x square centimeters. By substituting this variable into the formula, we can determine the length of the side.

We only consider the principal root because a polygon cannot have a negative side length. Now, since we are asked to write the answer using a rational exponent, we need to rewrite the radical expression.

As shown, we have found that the side length is x^(12) square centimeters.

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

Step 1

Step 2

Step 3

Step 4

Translating Between Rational Exponents and Radicals

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