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Chapter 5: Problem 97

Simplify each expression. See Examples 1 through 11. $$ (9 x y)^{2} $$

Short Answer

Expert verified
The simplified expression is \(81x^2y^2\).

Step by step solution

01

Apply the Power Rule

To simplify expressions with exponents, use the power rule. The power rule states that \[(a imes b)^n = a^n imes b^n.\] In this case, apply \[(9xy)^2\] which becomes \[9^2 imes x^2 imes y^2.\]
02

Calculate Each Power

Now, calculate the square of each component: \[9^2 = 81,\]\[x^2 = x imes x = x^2,\] and \[y^2 = y imes y = y^2.\]
03

Combine the Simplified Parts

Multiply these results to get the final simplified expression:\[81 imes x^2 imes y^2.\]Thus, the simplified expression is \[81x^2y^2.\]

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Key Concepts

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

Power Rule
When dealing with expressions that include exponents, the power rule is an essential tool. The power rule states:
  • If you have a product raised to a power, \[ (a \times b)^n = a^n \times b^n. \]
This means that you apply the exponent to each factor within the parenthesis. If you know how to distribute, think of this as distributing the exponent to each element inside.In the expression \( (9xy)^2 \), you use the power rule to expand it as:
  • The number: \( 9^2 \)
  • The variable \( x: x^2 \)
  • The variable \( y: y^2 \)
Exponents
Exponents are a shorthand way to show repeated multiplication of the same number. The exponent tells you how many times to multiply the base by itself. For instance, \( 9^2 \) means multiplying 9 by itself, which is 81.
  • The base is the number you repeatedly multiply.
  • The exponent is the small number written slightly above and to the right of the base.
When an exponent is applied to a variable, such as \( x^2 \), it means \( x \times x \). This is a fundamental concept in algebra because it shows how quantities scale together.Remember:
  • When raising a number to the power of 2, you are squaring it.
  • When raising a number to the power of 3, you are cubing it.
Multiplication of Powers
Multiplying powers with the same base means you add the exponents together. However, in this problem, you multiply the resultant powers after applying the power rule. For example:
  • The result of \(9^2 = 81\)
  • The result of \(x^2 = x \times x \)
  • The result of \(y^2 = y \times y \)
Now, we multiply these results to combine them fully: \[ 81 \times x^2 \times y^2 \]. The multiplication of powers in this case doesn't add the exponents on the x or y, because they were calculated separately. Instead, you're simply piecing together each part to form the complete simplified expression.

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Most popular questions from this chapter

Evaluate each expression using exponential rules. Write each result in standard notation. See Example 7. $$ \left(5 \times 10^{6}\right)\left(4 \times 10^{-8}\right) $$

Multiply. See Example 7. $$ (b+3)\left(2 b^{2}+b-3\right) $$

Perform the indicated operations. $$(3 x-1)(10 x-6)$$

Recall that the perimeter of a figure such as the ones shown in \(E x-\) ercises 73 through 76 is the sum of the lengths of its sides. Find the perimeter of each figure. A wooden beam is \(\left(4 y^{2}+4 y+1\right)\) meters long. If a piece \(\left(y^{2}-10\right)\) meters is cut, express the length of the remaining piece of beam as a polynomial in \(y .\) (GRAPH NOT COPY)

Simplify each expression. See Section \(5.1 .\) $$ \frac{x^{3} y^{6}}{x y^{2}} $$
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