Problem 147 Simplify each expression. $$ ... [FREE SOLUTION]

Chapter 5: Problem 147

Simplify each expression. $$ \left(8 a^{3}\right)^{2}(2 a)^{4} $$

Short Answer

Expert verified

1024a^{10}

Step by step solution

Apply the power of a power rule

The power of a power rule states that $\left(a^{m}\right)^{n} = a^{m \cdot n}$. Apply this rule to each term inside the parentheses: $\left(8a^3\right)^2 = 8^2 \cdot (a^3)^2 = 64a^6$ and $\left(2a\right)^4 = 2^4 \cdot a^4 = 16a^4$.

Multiply the numerical coefficients

Now multiply the numerical coefficients obtained in the first step: $64 \cdot 16 = 1024$.

Add the exponents of the same base

Combine the exponents for the variable $a$: $a^6 \cdot a^4 = a^{6+4} = a^{10}$.

Write the final answer

Combine the results from the previous steps to get the simplified expression: $1024a^{10}$.

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Key Concepts

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

Power of a Power Rule

First, let's understand the 'power of a power rule'. This rule in algebra helps us simplify expressions involving exponents. The rule states that when you have a power raised to another power, you simply multiply the exponents. For example, $\bigg(a^m\bigg)^n = a^{m \times n}$. Let鈥檚 break it down with the given exercise:
$\big(8a^3\big)^2 (2a)^4 $
According to the power of a power rule, we handle each part separately:

$\big(8a^3\big)^2$ turns into $8^2 \times a^{3 \times 2} = 64a^6$
$\big(2a\big)^4$ turns into $2^4 \times a^{1 \times 4} = 16a^4$

Now that these terms are simplified, we are ready for the next step.

Multiplication of Exponents

The next step involves combining the coefficients and the exponents. In algebra, you can multiply the numbers (coefficients) separately from the variables (with exponents). For the problem $\big(64a^6\big) (16a^4)$, first, multiply the numerical coefficients:
$64 \times 16$. This equals $1024$.
Multiplying the exponents is straightforward when the base is the same. Remember, the rule for multiplying exponents with the same base $a$ is to add their exponents: $a^m \times a^n = a^{m+n}$. In our case:

$a^6 \times a^4 = a^{6+4} = a^{10}$.

So, we have $1024a^{10}$.

Combining Like Terms

In simplifying expressions, combining like terms is essential, especially for variables with the same base. Like terms are terms with the same variable raised to the same power. Let's consider our final expression:
$1024a^{10}$
Because this term has only one variable and exponent, it is already in its simplest form, thus no further combination of like terms is necessary. However, in general, always ensure to:

Identify terms with the same variable and exponent
Add or subtract their coefficients as needed

This concept helps simplify more complex algebraic expressions effectively.

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91影视

Simplify each expression. $$ \left(8 a^{3}\right)^{2}(2 a)^{4} $$

Short Answer

Step by step solution

Apply the power of a power rule

Multiply the numerical coefficients

Add the exponents of the same base

Write the final answer

Key Concepts

Power of a Power Rule

Multiplication of Exponents

Combining Like Terms

One App. One Place for Learning.

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