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Magazine Chapter 1: Problem 42
Simplify the expression, writing your answer using positive exponents only. $$ \left(3 x^{-1}\right)^{2}\left(4 y^{-1}\right)^{3}(2 z)^{-2} $$
Short Answer
Expert verified
The simplified expression with positive exponents only is:
\[
\frac{144}{x^2 y^3 z^2}
\]
Step by step solution
01
Use Power of a Product Property
Apply the power of a product property, which states that \((ab)^n = a^n b^n\), to each term in the expression:
\[
(3x^{-1})^2 (4y^{-1})^3 (2z)^{-2}
\]
becomes:
\[
(3^2)(x^{-1})^2 (4^3)(y^{-1})^3 (2^{-2})(z^{-2})
\]
02
Simplify Exponential Terms
Now, compute each term with a numerical exponential:
\[
(9)(x^{-1})^2 (64)(y^{-1})^3 (2^{-2})(z^{-2})
\]
03
Apply Power of a Power Property
Use the power of a power property, which states that \((a^n)^m = a^{nm}\), to simplify the expression:
\[
(9)(x^{-2}) (64)(y^{-3}) (2^{-2})(z^{-2})
\]
04
Simplify Negative Exponents
To rewrite the expression with only positive exponents, remember that a value raised to a negative exponent is equivalent to its reciprocal raised to the positive exponent:
\[
(9)(x^{2})^{-1} (64)(y^{3})^{-1} (2^{-2})(z^{2})^{-1}
\]
Now, rewrite these as fractions:
\[
\frac{9}{x^2} \frac{64}{y^3} \frac{1}{2^2z^2}
\]
05
Multiply Fractions Together
Finally, multiply the fractions together by multiplying their numerators and denominators separately:
\[
\frac{9 \cdot 64 \cdot 1}{x^2 \cdot y^3 \cdot 2^2 \cdot z^2}
\]
06
Simplify the Fraction
Divide 9 and 64 by the greatest common divisor (1) and simplify:
\[
\frac{9 \cdot 64 \cdot 1}{x^2 \cdot y^3 \cdot 2^2 \cdot z^2} = \frac{576}{4x^2 y^3 z^2}
\]
The simplified expression with positive exponents only is:
\[
\frac{144}{x^2 y^3 z^2}
\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power of a Product Property
When simplifying expressions with exponents, a key tool to use is the **Power of a Product Property**. This property tells us how to handle expressions where both numbers and variables inside a product are raised to a power. The rule states that
For example, in the problem given, applying this property to \((3x^{-1})^2\) results in:
Using this property simplifies the expression into individual parts, making them ready for further simplification.
- \((ab)^n = a^n b^n\)
For example, in the problem given, applying this property to \((3x^{-1})^2\) results in:
- \(3^2 \cdot (x^{-1})^2\)
Using this property simplifies the expression into individual parts, making them ready for further simplification.
Power of a Power Property
Another critical concept is the **Power of a Power Property**. This property is applied when an exponential expression is raised to another power. The rule is written as:
Applying this to the expression \((x^{-1})^2\) in the problem changes it to:
Each part of the expression can be simplified using this property, enabling a straightforward approach to handle complex exponents in a step-by-step manner.
- \((a^n)^m = a^{nm}\)
Applying this to the expression \((x^{-1})^2\) in the problem changes it to:
- \(x^{-2}\)
Each part of the expression can be simplified using this property, enabling a straightforward approach to handle complex exponents in a step-by-step manner.
Negative Exponents
Negative exponents can often appear confusing, but they are actually quite simple to handle.
A negative exponent indicates that you take the reciprocal of the base. The fundamental rule is:
In the exercise, terms like \(x^{-2}\) and \(y^{-3}\) convert to:
A negative exponent indicates that you take the reciprocal of the base. The fundamental rule is:
- \(a^{-n} = \frac{1}{a^n}\)
In the exercise, terms like \(x^{-2}\) and \(y^{-3}\) convert to:
- \(\frac{1}{x^2}\)
- \(\frac{1}{y^3}\)
Fraction Simplification
Once you've expressed everything with positive exponents, the next step is **Fraction Simplification**. This involves multiplying and simplifying the resulting fractions.
Here's the approach:
Here's the approach:
- Multiply the numerators together.
- Multiply the denominators together.
- Simplify by dividing both by their greatest common divisor (GCD), if necessary.
- \(\frac{9}{x^2}\), \(\frac{64}{y^3}\), and \(\frac{1}{4z^2}\)
- \(\frac{9 \cdot 64 \cdot 1}{x^2 \cdot y^3 \cdot 4z^2} = \frac{576}{4x^2 y^3 z^2}\)
- \(\frac{144}{x^2 y^3 z^2}\)
