Question

Simplify each expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not \(0 .\) \(\frac{4 x^{-2}(y z)^{-1}}{2^{3} x^{4} y}\)

Short answer

\frac{1}{2 x^6 y^2 z}\

Step-by-step solution

- Simplify the constants in the fraction

Simplify the constants by dividing 4 by \(2^3\). \[ \frac{4}{2^3} = \frac{4}{8} = \frac{1}{2} \]

- Apply the negative exponents rule

Rewrite the negative exponents on the numerator as positive exponents in the denominator. \[ \frac{4 x^{-2}(y z)^{-1}}{2^3 x^4 y} = \frac{1}{2} \times \frac{1}{x^2 (yz)} \]

- Combine the exponents of the same base

Combine exponents of the same base in the denominator: \[ \frac{1}{2} \times \frac{1}{x^2 y z x^4 y} = \frac{1}{2} \times \frac{1}{x^{2+4} y^{1+1} z} = \frac{1}{2} \times \frac{1}{x^6 y^2 z} \]

- Simplify the final expression

Simplify the final expression to combine everything: \[ \frac{1}{2 x^6 y^2 z} \]

Key concepts

Negative Exponents

Negative exponents can seem tricky at first, but they follow a simple rule: any base with a negative exponent can be rewritten as its reciprocal with a positive exponent. For example, \(a^{-n}\) becomes \(\frac{1}{a^n}\). This helps to simplify expressions and make calculations easier. In our exercise, we had terms like \(x^{-2}\) and \((yz)^{-1}\) on the numerator. According to the rule, we can rewrite these as \(\frac{1}{x^2}\) and \(\frac{1}{yz}\) respectively, and move them to the denominator. By doing this, we can make all exponents positive, which simplifies the expression.

Fraction Simplification

Simplifying fractions involves reducing the numerator and the denominator to their simplest forms. First, look at the constants. In our case, we simplified \( \frac{4}{2^3} \). This was reduced to \( \frac{4}{8} = \frac{1}{2} \). Next, apply simplification rules for variables. For variables, remember to move terms with negative exponents to the other part of the fraction. We convert these variables so all exponents are positive. This makes it straightforward to combine and simplify further.

Algebraic Expressions

Algebraic expressions include terms with constants, variables, and exponents. Simplifying them is about applying rules and combining like terms. For example, in our solution, once we rewrote all terms with positive exponents, we then combined the exponents of the same base. In this case, \(x^2\) and \(x^4\) were combined to form \(x^{2+4} = x^6\). Similarly, we combined terms with the same base for \(y\), getting \(y^{1+1} = y^2\). Simplifying algebraic expressions requires careful application of these rules to achieve the final simplified form.