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Chapter 6: Problem 399

Simplify. $$\left(\frac{j^{2} j^{5}}{j^{4}}\right)^{3}$$

Short Answer

Expert verified
The simplified expression is \(j^9\).

Step by step solution

01

Simplify the inner expression

First, simplify the fraction inside the parentheses. Combine the exponents in the numerator using the rule for multiplication of exponents: ewline \ ewline \ ewline \(j^2 \times j^5 = j^{2+5} = j^7\)
02

Simplify the fraction

Next, simplify the fraction by applying the rule for division of exponents: ewline \ ewline \(\frac {j^7}{j^4} = j^{7-4} = j^3\)
03

Apply the outer exponent

Now, apply the exponent outside the parentheses using the power of a power rule: ewline \ ewline \((j^3)^3 = j^{3 \times 3} = j^9\)

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Key Concepts

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

Multiplication of Exponents
When you multiply numbers with the same base, you add their exponents. This is known as the multiplication of exponents. For example, in the problem, we have:

\(j^2 \times j^5 = j^{2+5} = j^7\)

This rule works because exponentiation is repeated multiplication. By adding the exponents, you're accounting for all the repeated multiplications.
  • Always combine the exponents by adding them when the bases are the same.

  • This rule simplifies many expressions and helps to solve more complex problems involving exponents.

Remember, the base must be the same for you to use this rule.
Division of Exponents
Just like you add exponents when multiplying, you subtract exponents when dividing numbers with the same base. In the example, we take the expression \(\frac{j^7}{j^4}\):

\(\frac{j^7}{j^4} = j^{7-4} = j^3\)

When dividing, you're distributing the repeated multiplication from the numerator to the denominator.
  • Ensure that both the numerator and denominator have the same base.

  • Subtract the exponent in the denominator from the exponent in the numerator.

This rule makes simplifying fractions with the same base straightforward.
Power of a Power Rule
The power of a power rule is used when you have an exponent raised to another exponent. In our problem, after simplification, we apply this rule to \((j^3)^3\):

\((j^3)^3 = j^{3 \times 3} = j^9\)

Here, you multiply the exponents to find the new exponent. This stems from the property of exponentiation that repeatedly applying an exponent is equivalent to multiplying the exponents.
  • Multiply the inner and outer exponents.

  • This rule is useful for simplifying and solving exponential expressions quickly.

By understanding and applying these rules, you can handle complex expressions involving exponents with ease.

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Most popular questions from this chapter

Mixed Practice (a) \(24 a^{5}+2 a^{5}\) (b) \(24 a^{5}-2 a^{5}\) (c) \(24 a^{5} \cdot 2 a^{5}\) (d) \(24 a^{5} \div 2 a^{5}\)

Simplify. (a) \(4^{-2}\) (b) \(10^{-3}\)

Divide each polynomial by the monomial. $$\left(63 m^{4}-42 m^{3}\right) \div\left(-7 m^{2}\right)$$

Divide the monomials. $$\frac{18 a^{4} b^{8}}{-27 a^{9} b^{5}}$$

Divide each polynomial by the monomial. $$\left(55 w^{2}-10 w\right) \div 5 w$$
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