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Chapter 6: Problem 391

Simplify. $$\left(\frac{a}{b^{6}}\right)^{3}$$

Short Answer

Expert verified
\(\frac{a^{3}}{b^{18}}\)

Step by step solution

01

Understand the Expression

The expression given is \(\bigg(\frac{a}{b^{6}}\bigg)^{3}\). It is a fraction raised to a power.
02

Apply the Power Rule

When raising a fraction to a power, apply the exponent to both the numerator and the denominator separately. This means \(\bigg(\frac{a}{b^{6}}\bigg)^{3} = \frac{a^{3}}{(b^{6})^{3}}\).
03

Simplify the Denominator

Now simplify the denominator by raising \(b^{6}\) to the power of \(3\). Use the power rule \((b^{m})^{n} = b^{m \times n}\). Thus, \((b^{6})^{3} = b^{6 \times 3} = b^{18}\).
04

Write the Simplified Expression

Combine the results from the numerator and the denominator. The expression simplifies to \(\frac{a^{3}}{b^{18}}\).

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Key Concepts

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

Exponent Rules
Exponent rules help us simplify expressions involving powers of variables.
These rules include the product rule, quotient rule, and power rule. They are essential for making algebra easier to manage. Understanding these rules helps you manipulate and simplify complex algebraic fractions.
For example:
Power Rule
One critical exponent rule is the Power Rule. It describes how to handle exponents when they are raised to another power.
The general rule is that \((a^{m})^n = a^{m \times n}\).
This means you multiply the exponents together.
In our exercise, we used the Power Rule: \[ (b^6)^3 = b^{6 \times 3} = b^{18} \] Applying the power rule correctly is key to simplifying and solving expressions.
Simplifying Expressions
To simplify means to make an expression easier to work with.
Use various exponent rules to break down and transform the original expression.
For instance, \[ \left( \frac{a}{b^6} \right)^3 \ \ \frac{a^3}{b^{18}} \] Here, simplify each part before combining again.
Algebraic Fractions
An algebraic fraction is a fraction where the numerator and/or the denominator are algebraic expressions.
This could include variables, exponents, and constants.
When simplifying algebraic fractions, apply exponent rules to both the numerator and the denominator.
Our example: \[ \left( \frac{a}{b^6} \right)^3 \ \ \frac{a^3}{b^{18}} \]
This involves handling variables with different powers at the same time.

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Most popular questions from this chapter

Divide each polynomial by the binomial. $$\left(64 x^{3}-27\right) \div(4 x-3)$$

Divide each polynomial by the binomial. $$\left(6 m^{2}-19 m-20\right) \div(m-4)$$

Divide each polynomial by the monomial. $$\frac{72 r^{5} s^{2}+132 r^{4} s^{3}-96 r^{3} s^{5}}{12 r^{2} s^{2}}$$

Divide each polynomial by the monomial. $$\frac{10 x^{2}+5 x-4}{-5 x}$$

Divide each polynomial by the binomial. $$\left(t^{2}-12 t+36\right) \div(t-6)$$
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