Problem 57 Simplify the expression. $$\le... [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 10: Problem 57

Simplify the expression. $$\left(2 p^{2}\right)^{3}(p q)^{2}$$

Short Answer

Expert verified

The simplified expression is \ \8 p^8 q^2 \.

Step by step solution

Identify the Base and Exponents

First, identify the base numbers and their exponents in the expression: \ $\left(2 p^{2}\right)^{3}(p q)^{2}$.

Apply the Power Rule on the First Term

Use the power rule $(a^m)^n = a^{m \times n}$ to simplify $(2p^2)^3$: Split it as follows: \ \left(2^3\right) \left(p^{2 \times 3}\right) \Rightarrow 8p^6\.

Apply the Power Rule on the Second Term

Next, simplify $(pq)^2$ using the power rule: Each term in the parentheses raised to the power individually, \ \Rightarrow p^2 \ q^2.\

Combine Simplified Parts

Multiply the simplified results from the previous steps: Combine $8p^6$ and $p^2 q^2$: \ 8 p^{6+2} q^2 = 8 p^8 q^2.\

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Key Concepts

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

Power Rule

In algebra, the power rule is fundamental for simplifying expressions with exponents. The power rule states that when you raise a power to another power, you multiply the exponents. This can be formulated as $(a^m)^n = a^{m \times n}$.

For example, in the original exercise, we simplified $(2p^2)^3$ by applying the power rule. First, we identify the base and the exponent inside the parentheses, which are 2 and $p^2$ respectively. Then, we raised both parts to the power of 3: $(2^3)(p^{2 \times 3}) = 8p^6$.

This step helps us break down complex expressions into simpler parts.

Exponents

Exponents are a way to denote repeated multiplication of a number by itself. The general notation is $a^n$, where $a$ is the base and $n$ is the exponent. This means $a$ is multiplied by itself $n$ times.

In our exercise, we saw different uses of exponents. For instance, in $(pq)^2$, both $p$ and $q$ are raised to the power of 2. By calculating each term separately, we get: $p^2q^2$.

Understanding exponents is crucial for simplifying and solving algebraic expressions.

Multiplication of Variables

When multiplying variables with exponents, you add the exponents if the bases are the same.

In the original exercise's final step, we combined the simplified parts $8p^6$ and $p^2q^2$. For the $p$ terms, we added the exponents: $p^6 \times p^2 = p^{6+2} = p^8$.

Finally, we multiplied $8p^8$ with $q^2$, resulting in $8p^8q^2$. This rule allows us to simplify expressions and keep them manageable.

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91影视

Simplify the expression. $$\left(2 p^{2}\right)^{3}(p q)^{2}$$

Short Answer

Step by step solution

Identify the Base and Exponents

Apply the Power Rule on the First Term

Apply the Power Rule on the Second Term

Combine Simplified Parts

Key Concepts

Power Rule

Exponents

Multiplication of Variables

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