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Chapter 2: Problem 91

Simplify the expression using one of the power rules. $$6(x y)^{3}$$

Short Answer

Expert verified
The simplified expression is \(6x^{3}y^{3}\).

Step by step solution

01

Identify the variables and the exponent

The given expression is: \(6(x y)^{3}\). Here, the variables are x and y, and the exponent is 3.
02

Apply the power rule to the variables

According to the power rule, when we have a product raised to a power, we can apply that power to each variable separately. So, \((xy)^{3} = x^{3}y^{3}\).
03

Write the simplified expression

Now that we have simplified the part with the variables, we can write the final expression: \(6(x y)^{3} = 6x^{3}y^{3}\). The simplified expression is \(6x^{3}y^{3}\).

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Key Concepts

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

Power Rule
The power rule is a handy tool in algebra that helps simplify expressions with exponents. It's especially useful when you have a product, like \((xy)^3\), raised to a power. The rule states that if you have a product of variables raised to a power, you can distribute that power to each variable separately.

Here's how it works:
- For the expression \((xy)^3\), you apply the exponent 3 to both variables: \(x^3\) and \(y^3\).
- This means \((xy)^3 = x^3y^3\).

In simple terms, the power rule allows each element in a product inside a parenthesis to get its own share of the exponent. Remember, this rule only applies when all elements inside the product are multiplied together.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are the foundation of algebra and allow us to represent mathematical ideas succinctly.

Let's break down the given expression: 6(xy)^3.

- Here, 6 is a coefficient, which is a constant multiplier of the variables.
- The term \((xy)^3\) consists of two variables, \(x\) and \(y\), inside parenthesis raised to an exponent.

When we simplify such expressions using rules like the power rule, we transform them into a more manageable form, like 6x^3y^3. This process is crucial for solving equations and understanding the relationships between numbers and variables.
Exponents
Exponents are a shorthand way to express repeated multiplication of the same number or expression. In the expression \((xy)^3\), the number "3" is the exponent. This tells us that \((xy)\) is to be multiplied by itself two more times:

- \((xy)^3 = (xy) \times (xy) \times (xy)\).

The exponent rule helps to simplify handling large numbers or variables repeatedly multiplied together. When applying the power rule to \((xy)^3\), the exponent is distributed to both \(x\) and \(y\), transforming the expression to \(x^3y^3\).

Exponents are powerful because they reduce complex multiplication into simpler terms, which helps streamline calculations and make algebraic manipulations easier.

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Most popular questions from this chapter

Write each number in scientific notation. A typical hard drive may hold approximately \(160,000,000,000\) bytes of data.

Simplify the expression using the product rule. Leave your answer in exponential form. $$\left(-7 t^{6}\right)\left(t^{3}\right)\left(-4 t^{7}\right)$$

Simplify the expression using one of the power rules. $$\left(\frac{t}{u}\right)^{10}$$

Simplify the expression using the product rule. Leave your answer in exponential form. $$8 y^{3} \cdot y^{2}$$

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{8 \times 10^{7}}{4 \times 10^{4}}\)
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