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Chapter 5: Problem 28

Express using positive exponents and, if possible, simplify. $$\frac{y^{4} z^{-3}}{7 x^{-2}}$$

Short Answer

Expert verified
\(\frac{y^4 x^2}{7z^3}\)

Step by step solution

01

Understand the problem

The given expression is \(\frac{y^{4} z^{-3}}{7 x^{-2}}\). The task is to express it using positive exponents and simplify if possible.
02

Rewrite using positive exponents

To change negative exponents to positive exponents, use the property \(a^{-n} = \frac{1}{a^n}\). Apply this to the expression: \(z^{-3} = \frac{1}{z^3}\) and \(x^{-2} = \frac{1}{x^2}\).
03

Simplify the expression

Rewrite the given expression: \(\frac{y^{4} z^{-3}}{7 x^{-2}} = \frac{y^{4}}{7 x^{-2} z^{3}}\). Apply the property to \(x^{-2}\), which becomes \(x^2\) in the numerator. The expression now is \(\frac{y^{4} x^{2}}{7 z^{3}}\).

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Key Concepts

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

Fraction Simplification
Simplifying fractions is a key skill in algebra. When dealing with algebraic fractions, the numerator (top part) and denominator (bottom part) can both contain variables and exponents. To simplify fractions, you can often break down both parts to their simplest forms, cancel out common factors, and make sure all exponents are positive. For example, in the expression \(\frac{y^{4} z^{-3}}{7 x^{-2}}\), the goal is to rewrite it in the simplest form and ensure all exponents are positive.
Negative Exponents
Negative exponents tell you to take the reciprocal (invert the fraction) and change the exponent to positive. The property \(a^{-n} = \frac{1}{a^n}\) is very useful here. For instance, in the expression \(z^{-3}\), you can rewrite it as \(\frac{1}{z^{3}}\). Similarly, \(x^{-2}\) becomes \(\frac{1}{x^{2}}\). By applying this property, we can convert all the negative exponents in our initial expression to positive exponents.
Algebraic Expressions
Algebraic expressions consist of variables, exponents, and coefficients combined using mathematical operations. In the expression \(\frac{y^{4} z^{-3}}{7 x^{-2}}\), \(y\), \(z\), and \(x\) are variables, while 4, -3, and -2 are the exponents. The constant 7 is the coefficient in the denominator. When working with algebraic expressions, it's crucial to understand how to manipulate these components. By applying the rules of exponents and simplifying the fractions, we can often rewrite complex expressions in a simpler, more manageable form. For example, by simplifying the given expression, it becomes \(\frac{y^{4} x^{2}}{7 z^{3}}\), where all exponents are positive.

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Most popular questions from this chapter

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{7 x}{x-2}\\\ &g(x)=3 x+7 \end{aligned} $$

If \(f(x)=c,\) where \(c\) is some positive constant, describe how the graphs of \(y=g(x)\) and \(y=(f+g)(x)\) will differ.

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{22} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{a^{4}}{b^{3}}\right)^{5} $$

Interest Compounded Annually. An amount of money \(P\) that is invested at the yearly interest rate r grows to the amount $$ P(1+r)^{t} $$ after \(t\) years. Find a polynomial that can be used to determine the amount to which \(P\) will grow after 2 years.
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