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Chapter 10: Problem 263

In the following exercises, simplify. $$\left(\frac{a^{4} \cdot a^{6}}{a^{3}}\right)^{2}$$

Short Answer

Expert verified
The simplified expression is $$a^{14}.$$

Step by step solution

01

Simplify the expression inside the parentheses

First, simplify the expression inside the parentheses by using the properties of exponents. Recall that when multiplying exponents with the same base, you add the exponents: $$a^{4} \times a^{6} = a^{4+6} = a^{10}$$. This gives: $$\frac{a^{10}}{a^{3}}.$$
02

Simplify the division inside the parentheses

Next, simplify the fraction inside the parentheses using the properties of exponents. When dividing powers with the same base, you subtract the exponents. $$a^{10} \text{ divided by } a^{3} = a^{10-3} = a^{7}.$$
03

Apply the outer exponent

After simplifying the expression inside the parentheses, apply the outer exponent. Recall the power of a power rule: $$(a^{m})^{n} = a^{m \times n}$$ This gives: $$\big(a^{7}\big)^{2} = a^{7 \times 2} = a^{14}.$$

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Key Concepts

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

Properties of Exponents
Exponents are a way to express repeated multiplication of the same number. There are some properties of exponents that we use to simplify expressions. By understanding these rules, you can tackle any problem involving exponents with ease.
Here are the basic properties:
  • Product of Powers: When multiplying two expressions with the same base, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers: When dividing two expressions with the same base, subtract the exponents. For example, \(\frac{a^m}{a^n} = a^{m-n}\).
  • Power of a Power: When raising an exponent to another exponent, multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).

Knowing these properties is crucial for simplifying complex expressions, as we will see in the following sections.
Multiplication of Exponents
Multiplication of exponents involves using the product of powers property. This property states that when you have the same base, you can add the exponents together.
Let鈥檚 start by looking at an example from our exercise: we have \(a^4 \cdot a^6\).
According to the product of powers rule, this can be simplified to:
\[ a^4 \cdot a^6 = a^{4+6} = a^{10} \]
This step significantly reduces the complexity of the expression, making it easier to handle the rest of the problem.
Division of Exponents
When dividing exponents with the same base, you use the quotient of powers property. This property states that you should subtract the exponent of the denominator from the exponent of the numerator.
Continuing with our example, after simplifying the multiplication, we have \(\frac{a^{10}}{a^3}\).
Applying the quotient of powers rule, we get:
\[ \frac{a^{10}}{a^3} = a^{10-3} = a^7 \]
This step helps simplify our expression further.
Power of a Power Rule
The power of a power rule is another crucial property. This rule is applied when you have one exponent raised to another. You simply multiply the exponents.
Finally, using our simplified expression \(a^7\), we apply the outer exponent (2) to get:
\[ \big(a^7\big)^2 = a^{7 \times 2} = a^{14} \]
Combining all the steps, we simplif:ed \(\big(\frac{a^4 \cdot a^6}{a^3}\big)^2\) to \(a^{14}\).
By understanding and applying the properties of exponents, we can simplify even the most complex expressions with ease.

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Most popular questions from this chapter

Convert each number to decimal form. $$2.9 \times 10^{4}$$

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