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Chapter 10: Problem 4

Perform the indicated operations. $$\left(x^{4} y^{3}\right)^{3}$$

Short Answer

Expert verified
The final expression is \(x^{12} y^9\).

Step by step solution

01

Understand the Power Rule

When raising a power to another power, multiply the exponents. This means for an expression \((a^m)^n\), the result is \((a)^{m \cdot n}\).
02

Apply the Power Rule to Each Variable

Apply the rule to \(x^4\) and \(y^3\) separately: \((x^4)^3\) and \((y^3)^3\).
03

Perform the Multiplication for each Variable

For \((x^4)^3\), multiply the exponents: \4 \times 3 = 12\. Thus, \((x^4)^3 = x^{12}\). Then, for \((y^3)^3\), multiply the exponents: \3 \times 3 = 9\. Thus, \((y^3)^3 = y^9\).
04

Combine the results

Combine the results of both variables: \(x^{12} \cdot y^9\).

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Key Concepts

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

Power Rule
In mathematics, the power rule is a key concept when dealing with exponents. It states that when you raise a power to another power, you multiply the exponents.

Think of it as \((a^m)^n\) becomes \(a^{m \times n}\).
In simpler terms, if you see an exponentiation inside another exponentiation, you just multiply the exponents together.

For instance:
  • If you have \((x^5)^2\), that turns into \((x)^{5 \times 2} = x^{10}\).
This rule helps a lot with simplifying more complex expressions. Let's apply it to our exercise!
Multiplying Exponents
Using the power rule requires us to multiply the exponents. It's straightforward and simply involves performing basic multiplication.

Consider our example. In the expression \((x^4)^3\), the exponent 4 is raised to the exponent 3. By the power rule, you multiply 4 by 3:

\(4 \times 3 = 12\).
Therefore, \((x^4)^3 = x^{12}\).
Similarly, for the \((y^3)^3\) part:

\(3 \times 3 = 9\),
so \((y^3)^3 = y^9\).
Variables in Exponents
When working with variables in exponents, you follow the same rules as you do with numbers.

Consider variables like x and y in our exercise. When we apply the power rule and multiply their exponents, we focus on the exponent values and the variables remain unchanged.

Using the exercise:
  • For \((x^4)^3\), apply the rule: multiply the exponents 4 and 3 to get \((x)^{4 \times 3} = x^{12}\).
  • Next, for \((y^3)^3\), do the same: \((y)^{3 \times 3} = y^9\).
Remember, the variables themselves do not change; just their exponents are multiplied.
Combining Exponential Terms
After applying the power rule and calculating the new exponents, the last step is combining the exponential terms.

From our solution steps, we simplified \((x^4)^3\) to \(x^{12}\), and \((y^3)^3\) to \(y^9\).

Now, combine these results:

\(x^{12} \times y^9\).

This is the final simplified form. To recall:
  • Always apply the power rule first.
  • Multiply exponents for each variable.
  • Combine the results to get your final answer.
With practice, these steps will become second nature!

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Most popular questions from this chapter

Simplify the expression. $$\left(6 a^{2} b\right)^{2}$$

Subtract \(\left(9 x^{2}+16 x-4\right)\) from \(\left(2 x^{2}-4 x-9\right)\)

Fill in the missing polynomial. $$(x+3)(\quad)=x^{2}+8 x+15$$

Simplify. Write the answers with positive exponents only. $$\left(4 m^{3} n^{-5}\right)\left(m^{-4} n^{6}\right)$$

Perform the indicated operations. $$\left(\frac{7}{8} a-4 b-\frac{1}{2} c\right)-\left(-\frac{1}{4} a+\frac{1}{2} b+\frac{3}{8} c\right)$$
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