Problem 15 Simplify. $$\left(-\frac{3}{2}... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 4: Problem 15

Simplify. $$\left(-\frac{3}{2}\right)^{3}$$

Short Answer

Expert verified

-\frac{27}{8}

Step by step solution

Understand the Exponent

The given expression is $\textstyle \big(-\frac{3}{2}\big)^3 $. This means $-\frac{3}{2}$ is multiplied by itself three times.

Multiply the Fraction

To simplify $\big(-\frac{3}{2}\big)^3 $, multiply $-\frac{3}{2}$ by itself three times: $\big(-\frac{3}{2}\big) \times \big(-\frac{3}{2}\big) \times \big(-\frac{3}{2}\big) $.

Calculate Step-by-Step

First, multiply $-\frac{3}{2} $ by itself: $\big(-\frac{3}{2}\big) \times \big(-\frac{3}{2}\big) = \frac{9}{4} $ because multiplying two negatives gives a positive. Next, multiply the result by $-\frac{3}{2}$: $\frac{9}{4} \times -\frac{3}{2} = -\frac{27}{8} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Multiplication

When you multiply fractions, you multiply the numerators together and then the denominators together. In the problem $\big(-\frac{3}{2}\big)^3$, the fraction $-\frac{3}{2}$ is repeated three times.

First computation: $-\frac{3}{2} \times -\frac{3}{2} = \frac{9}{4}$.
This works because multiplying two negative numbers results in a positive product.
Second computation: \frac{9}{4} \times -\frac{3}{2} = -\frac{27}{8}\.
This results in a negative number since one of the factors is negative.

It's essential to correctly handle the negative signs and make sure you simplify whenever possible.

Negative Exponents

When dealing with negative exponents, remember these key points:

A negative exponent indicates that the base should be reciprocated (flipped) before applying the exponent.
For example, $x^{-n} = \frac{1}{x^n}$.
In our original problem, $-\frac{3}{2}$ raised to the positive power of 3 doesn't involve a negative exponent, but knowing how to deal with them is essential for more complex problems.

Understanding how to handle these will make simplifying expressions much easier.

Simplifying Exponents

When simplifying expressions with exponents, follow these guidelines:

Apply the exponent to both the numerator and the denominator.
For the given problem, \[\big(-\frac{3}{2}\big)^3\] means multiplying $-\frac{3}{2}$ three times.
Start by multiplying two fractions to find an intermediate result.
Finally, multiply the intermediate result by the fraction one more time.

Use these techniques, and you'll be able to simplify exponent expressions accurately.

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91影视

Simplify. $$\left(-\frac{3}{2}\right)^{3}$$

Short Answer

Step by step solution

Understand the Exponent

Multiply the Fraction

Calculate Step-by-Step

Key Concepts

Fraction Multiplication

Negative Exponents

Simplifying Exponents

One App. One Place for Learning.

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