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Chapter 1: Problem 75

Simplify. $$ \frac{\left(9 x y^{3} z\right)^{2}}{3(x y z)^{2}} $$

Short Answer

Expert verified
The simplified expression is \(27y^4\).

Step by step solution

01

Expand the Numerator

First, expand the numerator. We have \((9xy^3z)^2\). Applying the power to each term, we get: \(9^2 \cdot x^2 \cdot (y^3)^2 \cdot z^2\). Thus, it becomes: \(81x^2y^6z^2\).
02

Expand the Denominator

Next, expand the denominator \(3(xyz)^2\). First calculate \((xyz)^2 = x^2y^2z^2\), and then multiply by 3: \(3x^2y^2z^2\).
03

Simplify the Fraction

Now, substitute the expanded forms into the fraction: \(\frac{81x^2y^6z^2}{3x^2y^2z^2}\). Simplify by dividing the corresponding terms: \(\frac{81}{3} = 27\), \(\frac{x^2}{x^2} = 1\), \(\frac{y^6}{y^2} = y^4\), \(\frac{z^2}{z^2} = 1\). Therefore, the result is \(27y^4\).

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Key Concepts

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

Power of a Product
When dealing with expressions such as \((9xy^3z)^2\), understanding the "power of a product" rule is crucial. This rule states that to raise a product to a power, you must apply the exponent to each element within the parentheses. For example, \((ab)^n\) means you apply the power to both 'a' and 'b', resulting in \(a^n \cdot b^n\).

Let's break down \((9xy^3z)^2\):
  • First, square the constant: \(9^2 = 81\).
  • Next, apply the power to the variable 'x': \(x^2\).
  • For the power of 'y', remember \((y^m)^n = y^{mn}\): \((y^3)^2 = y^6\).
  • Finally, apply the power to 'z': \(z^2\).
Putting it all together, you get \(81x^2y^6z^2\). Using the power of a product efficiently simplifies working with complex algebraic expressions.
Fraction Simplification
Simplifying fractions is a key algebraic skill that involves reducing a fraction to its simplest form. In the case of the fraction \(\frac{81x^2y^6z^2}{3x^2y^2z^2}\), the goal is to divide the numerator by the denominator, simplifying as much as possible.
  • Numerical Coefficients: Divide 81 by 3, and you will get 27.
  • Variable 'x': \(x^2\) divided by itself gives 1.
  • Variable 'y': Use the exponent subtraction rule: \(y^6 - y^2 = y^4\).
  • Variable 'z': \(z^2\) divided by itself also equals 1.
After simplifying, the expression boils down to just \(27y^4\). The fraction is significantly reduced, making further calculations easier and the expression more understandable.
Exponents in Algebra
Exponents in algebra are a way of expressing repeated multiplication. They are pivotal in simplifying expressions and play a critical role in reducing complex terms. In our example, various exponent rules are applied, which are essential to handle algebraic expressions.
  • Power of a Power Rule: As seen with \((y^3)^2 = y^6\), applying an exponent to another exponent multiplies the exponents.
  • Multiplication of Like Bases: When multiplying like bases, such as in \(x^2 \cdot x^2\), the exponents add, yielding \(x^{2+2} = x^4\).
  • Division of Like Bases: The rule \(a^m/a^n = a^{m-n}\) allows us to simplify expressions like \(y^6/y^2 = y^{6-2} = y^4\).
By mastering these rules, students can handle more complex algebraic expressions confidently. Exponents streamline mathematical notation, enabling the simplification of otherwise cumbersome calculations.

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Most popular questions from this chapter

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=\sqrt{x} ; g(x)=x^{3}-1 $$

How do two graphs differ if their functions are the same except that the domain of one excludes some \(x\) -values from the domain of the other?

For each function, find and simplify \(f(x+h)\). $$ f(x)=3 x^{2} $$

Find, rounding to five decimal places: a. \(\left(1+\frac{1}{100}\right)^{100}\) b. \(\left(1+\frac{1}{10,000}\right)^{10,000}\) c. \(\left(1+\frac{1}{1,000,000}\right)^{1,000,000}\) d. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

How do the graphs of \(f(x)\) and \(f(x+10)\) differ?
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