Problem 49 Simplify the following expressio... [FREE SOLUTION]

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Prealgebra

Charles P. McKeague

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 9: Problem 49

Simplify the following expressions. $\left(a^{3}\right)^{7} \cdot\left(a^{4}\right)^{2}$

Short Answer

Expert verified

The simplified expression is $a^{29}$.

Step by step solution

Apply the Power of a Power Rule

To simplify the expression, first apply the rule $(x^m)^n = x^{m\cdot n}$ separately to each part of the expression. For the first part, $(a^3)^7 = a^{3\cdot7} = a^{21}$. For the second part, $(a^4)^2 = a^{4\cdot2} = a^8$.

Apply the Product of Powers Rule

Now, the expression becomes $a^{21} \cdot a^8$. Apply the rule for the product of the same base: $x^m \cdot x^n = x^{m+n}$. Add the exponents together: $a^{21+8} = a^{29}$.

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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

When you encounter an expression where an exponent is raised to another exponent, you'll want to use the Power of a Power Rule. This rule tells us that when you have a power that is raised to another power, you can simplify it by multiplying the exponents. In mathematical terms, the rule is expressed as

$(x^m)^n = x^{m \cdot n}$.

This means that if you have

$(a^3)^7$, the exponents 3 and 7 are multiplied to give $a^{21}$.
Similarly, for $(a^4)^2$, you multiply 4 by 2 to get $a^8$.

Remember, the Power of a Power Rule helps you turn complex climbs into simple steps by multiplying the exponents. This concept is often one of the first steps toward simplifying exponential expressions.

Product of Powers Rule

After you've used the Power of a Power Rule to simplify separate parts of your expression, it's time to combine them using the Product of Powers Rule. This rule is straightforward when you're working with terms that have the same base. It states:

$x^m \cdot x^n = x^{m+n}$.

In simpler terms, when you are multiplying like bases, you can add the exponents together. In our example, once we have

$a^{21} \cdot a^8$, you add the 21 and 8 to get $a^{29}$.

This rule simplifies expressions by allowing you to reduce multiple powers into one single, larger power. It's a powerful tool for quickly working through problems involving exponential operations.

Simplifying Expressions

Simplifying expressions is like tidying up a room鈥攅verything should be neat and concise. The goal is to express the problem in its most straightforward form. For exponential expressions,

we aim to collect them into a single term whenever possible.

Let's take an expression such as

$(a^3)^7 \cdot (a^4)^2$.

By applying our known rules, we end up with the simplified form

$a^{29}$.

The process involves sequentially applying the Power of a Power and the Product of Powers rules, which helps break down the complex into something simpler and more manageable.

Always aim to express your final answer with the fewest possible terms and the simplest form.

This approach not only makes the expressions easier to read but also aids in further mathematical operations or solving other problems.

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91影视

Simplify the following expressions. \(\left(a^{3}\right)^{7} \cdot\left(a^{4}\right)^{2}\)

Short Answer

Step by step solution

Apply the Power of a Power Rule

Apply the Product of Powers Rule

Key Concepts

Power of a Power Rule

Product of Powers Rule

Simplifying Expressions

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