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Chapter 10: Problem 10

Simplify. $$\left(\frac{5 c^{2}}{d}\right)^{3}$$

Short Answer

Expert verified
\(\frac{125 c^6}{d^3}\)

Step by step solution

01

Understand the expression

The given expression is \(\bigg(\frac{5 c^2}{d}\bigg)^3\). This means we need to cube both the numerator and the denominator separately.
02

Cube the numerator

Cube the numerator \(5 c^2\): \((5 c^2)^3 = 5^3 \times (c^2)^3\). Simplify this to \(125 c^6\).
03

Cube the denominator

Cube the denominator \(d\): \((d)^3 = d^3\).
04

Write the simplified expression

Combine the cubed numerator and denominator to get the simplified expression: \(\frac{125 c^6}{d^3}\).

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Key Concepts

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

Cubing Expressions
When you see an expression like \(\bigg( \frac{5 c^2}{d}\bigg)^3\), it means you are raising both the numerator and the denominator to the power of 3.
This is because cubing an entire fraction is the same as cubing its parts.
  • First, let鈥檚 break down the numerator and the denominator.
  • The numerator is \( 5c^2 \) and the denominator is \( d \).

Now, when we cube \(5c^2\), we need to cube both 5 and \(c^2\):
  • \( (5 c^2)^3 = 5^3 \times (c^2)^3 \) which simplifies to \(125 c^6\).
Similarly, we cube the denominator \(d\) to get:
  • \( d^3 \).

Therefore, the cubed expression becomes \( \frac{125 c^6}{d^3} \).
The most important thing to remember when cubing an expression is to apply the exponent to both the base and the exponent of each term inside the expression.
Exponents in Algebra
Exponents are a way of indicating repeated multiplication. For example, \( c^2 \) means \( c \times c \).
When you have an expression with exponents, you follow the power rules.
  • \( (a^m)^n = a^{m\times n} \).
This means if you have \( (c^2)^3 \), you multiply the exponents:
  • \( c^{2\times 3} = c^6 \).
This is why the problem above simplifies \( (c^2)^3 \) to \( c^6 \).
It鈥檚 useful to know the other exponent rules, like:
  • \( a^m \times a^n = a^{m+n} \)
  • \( \frac{a^m}{a^n} = a^{m-n} \)
  • \( (ab)^m = a^m \times b^m \)

Understanding and mastering these rules is fundamental in simplifying and solving algebraic expressions efficiently.
Fractional Expressions
Fractional expressions involve both a numerator and a denominator. Simplifying these can sometimes seem tricky, but the steps are straightforward.

Let鈥檚 take a look at our expression: \( \bigg( \frac{5 c^2}{d}\bigg)^3 \). To simplify, we cube both the numerator and the denominator:
  • Cube the numerator \( (5 c^2)^3 \) to get \( 125 c^6 \).
  • Cube the denominator \( d^3 \).

This results in \( \frac{125 c^6}{d^3} \).
Another trick to remembering how to simplify fractional expressions is to visualize each part separately, cube them, and then combine them back.

If you remember to apply the cube to both parts of your fraction, fractional expressions become much easier to handle.
Practicing with multiple examples can help reinforce this concept.

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Most popular questions from this chapter

Perform the indicated operations. $$\left(2 x^{3}-5 x+8\right)-\left(4 x^{2}+2 x-3\right)+\left(-7 x^{3}+8 x^{2}\right)$$

Perform the indicated operations. $$\left(-9 m^{4}-7 m^{2}+8\right)-\left(3 m^{4}+9 m^{3}-11\right)-\left(8 m^{3}+m^{2}-3\right)$$

Simplify the expression. $$\left(\frac{2 w}{p}\right)^{4}$$

Simplify the expression. $$\left(\frac{9 u}{t}\right)^{2}$$

Multiply the polynomials. $$\begin{array}{r}10 c^{2}+3 c-6 \\\\\times \quad 2 c^{2}+c+2 \\\\\hline\end{array}$$
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