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Chapter 10: Problem 7

Simplify. $$\left(-4 c d^{5}\right)^{2}$$

Short Answer

Expert verified
\(\big(-4cd^{5}\big)^{2}= 16c^{2}d^{10}\)

Step by step solution

01

Understand the Problem

The expression \(\big(-4cd^{5}\big)^{2} \) needs to be simplified. This involves squaring both the coefficient \(-4\) and the variable part \(cd^{5}\).
02

Square the Coefficient

Square \(-4\). \(\big(-4\big)^2 = 16 \). So, the coefficient part will be 16.
03

Square the Variable Part

Square both \(c\) and \(d^{5}\). When \(c\) is squared, it becomes \(c^{2}\), and when \(d^{5}\) is squared, it becomes \(d^{10}\).
04

Combine the Squared Terms

The squared coefficient and variables together give \(\big(16 \cdot c^2 \cdot d^{10}\big)\).

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Key Concepts

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

Squaring Coefficients
When you square a coefficient, you multiply it by itself. This might sound simple, but it鈥檚 crucial to avoid errors. For instance, in the expression \(-4\)^{2},\ we multiply -4 by -4. Recall that multiplying two negative numbers results in a positive number. Hence, \(-4 \cdot -4 = 16\). Squaring is fundamental in algebra. Always remember this rule.
Exponents
Exponents signify repeated multiplication. For example, \(d^{5}\) means multiplying the variable \(d\) by itself five times: \(d \cdot d \cdot d \cdot d \cdot d\). When you square an expression with an exponent, multiply the exponents. In \((cd^{5})^2\), the \(c\) and \(d^{5}\) terms are squared. This transforms \(d^{5} \cdot d^{5} \) into \(d^{10}\), following the rule that \((d^{5})^{2} = d^{5 \cdot 2} = d^{10}\). Always apply these rules to simplify expressions.
Algebraic Expressions
Algebraic expressions consist of variables, coefficients, and exponents. Combining them correctly is critical. For \((-4cd^{5})^{2}\), follow these steps:
  • Square each part separately.
  • Square the coefficient: \(-4\)^{2}=16\.
  • Square the variables: \((c^{1})^{2} = c^{2}\) and \((d^{5})^{2} = d^{10}\).
  • Combine the results: 16c^{2}d^{10}\.
Always recheck each step to avoid mistakes. Simplifying involves following these straightforward rules correctly.

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Most popular questions from this chapter

Simplify the expression. $$(3 t)^{2}$$

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{-2}$$

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