Problem 43
Question
For the following exercises, simplify the given expression. Write answers with positive exponents. $$\left(9 z^{3}\right)^{-2} y$$
Step-by-Step Solution
Verified Answer
The expression simplifies to \( \frac{y}{81z^{6}} \).
1Step 1: Understand the Problem
The expression given is \( (9z^{3})^{-2}y \). The goal is to simplify this expression and ensure no negative exponents remain in the final answer.
2Step 2: Apply Negative Exponent Rule
The negative exponent rule states that \( a^{-b} = \frac{1}{a^{b}} \). Applying this rule to the expression \( (9z^{3})^{-2} \) gives us \( \frac{1}{(9z^{3})^{2}} \).
3Step 3: Simplify the Inside of the Parenthesis
Simplify \((9z^{3})^{2}\) by applying the power of a product rule \((ab)^{c} = a^{c}b^{c}\). This results in \(9^{2}z^{6}\).
4Step 4: Substitute Back into the Expression
Place the simplified form back into the expression: \( \frac{1}{9^{2}z^{6}}y \).
5Step 5: Simplify the Expression
Compute \(9^2\) which is \(81\). So, the expression becomes \( \frac{y}{81z^{6}} \). The final expression has all positive exponents.
Key Concepts
Negative ExponentsPower of a Product RuleSimplifying Expressions
Negative Exponents
Negative exponents can be a little tricky at first, but once you understand the basic rule, they're easy to handle. The rule for negative exponents is that any base with a negative exponent, say \( a^{-b} \), equals the reciprocal of the base with a positive exponent, which is \( \frac{1}{a^b} \). In other words, a negative exponent tells us to "flip" the base into a fraction.
Let's look at an example: if you have \( (9z^3)^{-2} \), it means you take the reciprocal and change the sign of the exponent: \( \frac{1}{(9z^3)^2} \).
By understanding this rule, negative exponents become less daunting, allowing you to rewrite expressions with positive exponents, which are generally easier to work with.
Let's look at an example: if you have \( (9z^3)^{-2} \), it means you take the reciprocal and change the sign of the exponent: \( \frac{1}{(9z^3)^2} \).
By understanding this rule, negative exponents become less daunting, allowing you to rewrite expressions with positive exponents, which are generally easier to work with.
Power of a Product Rule
The power of a product rule is another important concept in algebra. It helps us simplify expressions where a product, such as \((ab)^c\), is raised to a power. This rule states that \((ab)^c = a^c \times b^c\).
In simpler terms, you raise each part of the product to the power separately.
Consider \((9z^3)^{2}\) from our original problem. Using the rule, you break it into \(9^2 \times (z^3)^2\). This results in \(9^2 \times z^{6}\).
This rule is a powerful tool that makes complex expression handling more straightforward by "distributing" the exponent over each factor inside the parentheses.
In simpler terms, you raise each part of the product to the power separately.
Consider \((9z^3)^{2}\) from our original problem. Using the rule, you break it into \(9^2 \times (z^3)^2\). This results in \(9^2 \times z^{6}\).
This rule is a powerful tool that makes complex expression handling more straightforward by "distributing" the exponent over each factor inside the parentheses.
Simplifying Expressions
Simplifying expressions is like tidying up a math equation so it's clearer and easier to read. It involves reducing the expression to its simplest form without changing its value or meaning.
For example, in the expression \(\frac{1}{9^2z^6}y\), we simplify by calculating \(9^2\), which is 81. This gives the simpler form \(\frac{y}{81z^6}\).
Simplifying is not just about solving the problem more efficiently; it makes complex expressions much more manageable and easier to work with.
For example, in the expression \(\frac{1}{9^2z^6}y\), we simplify by calculating \(9^2\), which is 81. This gives the simpler form \(\frac{y}{81z^6}\).
- Always perform any operations like powers or multiplications before simplifying.
- Look for ways to write fractions in simpler terms.
- Ensure no negative exponents are left in the final result unless instructed otherwise.
Simplifying is not just about solving the problem more efficiently; it makes complex expressions much more manageable and easier to work with.
Other exercises in this chapter
Problem 43
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Simplify each expression. $$\sqrt{\frac{225 x^{3}}{49 x}}$$
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For the following exercises, simplify the expression. $$ 7 z-3+z \cdot 6^{2} $$
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For the following exercises, factor the polynomials. $$ 4 x(x-1)^{-\frac{2}{3}}+3(x-1)^{\frac{1}{3}} $$
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