Problem 573 Simplify. $$\left(\frac{x}{2}\... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 573

Simplify. $$\left(\frac{x}{2}\right)^{5}$$

Short Answer

Expert verified

\frac{x^5}{32}

Step by step solution

Understand the Expression

The given expression is $\frac{x}{2}\right)^{5}$, which means we need to raise both the numerator and the denominator to the power of 5.

Apply the Power to the Numerator and Denominator Separately

Raise the numerator and the denominator separately to the 5th power. So, we have $\frac{x}{2}\right)^{5} = \frac{x^5}{2^5}$.

Simplify the Denominator

Calculate 2 to the power of 5: \[ 2^5 = 32 \].

Write the Simplified Expression

Combine the simplified numerator and denominator to get the final expression: \[ \frac{x^5}{32} \].

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Key Concepts

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

raising to a power

Raising a number or variable to a power means multiplying that number or variable by itself a certain number of times. For example, raising a number to the power of 5 means we multiply it by itself 5 times. We use the notation $ x^5 $ to represent raising x to the 5th power. In the exercise $ \left( \frac{ x }{ 2 } \right)^{ 5 } $, we are raising both the numerator (x) and the denominator (2) to the 5th power. This gives us \left( \frac{ x }{ 2 } \right)^{ 5 } = \frac{ x^5 }{ 2^5 }\.

numerator and denominator

In a fraction, the numerator is the number on the top, and the denominator is the number on the bottom. The numerator represents parts of the whole, while the denominator represents the total number of equal parts. For example, in the fraction $ \frac{ 1 }{ 2 } $, 1 is the numerator, and 2 is the denominator. In the given exercise, $ \left( \frac{ x }{ 2 } \right)^{ 5 } $, x is the numerator, and 2 is the denominator. When we raise the fraction to a power, we apply the power to both the numerator and the denominator separately, as seen in \frac{ x^5 }{ 2^5 }\.

simplifying fractions

Simplifying fractions means reducing them to their simplest form. To do this, we divide the numerator and the denominator by their greatest common divisor (GCD). However, in expressions where we are working with powers, the process involves calculating the power first before checking if further simplification is needed. In our problem, after raising both numerator and denominator to the power of 5, we get \frac{ x^5 }{ 2^5 } = \frac { x^5 }{ 32 }\, given that 2 raised to the power of 5 is 32. Since no common factors exist between $ x^5 $ and 32, this fraction is already in its simplest form.

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91影视

Simplify. $$\left(\frac{x}{2}\right)^{5}$$

Short Answer

Step by step solution

Understand the Expression

Apply the Power to the Numerator and Denominator Separately

Simplify the Denominator

Write the Simplified Expression

Key Concepts

raising to a power

numerator and denominator

simplifying fractions

One App. One Place for Learning.

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