Question

Evaluate the following expressions for \(x=2, y=-3,\) and \(z=-5\) $$ \left(\frac{x}{z}\right)^{-y} $$

Short answer

Answer: \(\frac{-125}{8}\)

Step-by-step solution

Substituting the values of \(x, y\) and \(z\) into the expression

Replace the variables \(x\), \(y\), and \(z\) in the expression with their given values (\(x=2, y=-3,\) and \(z=-5\)): $$ \left(\frac{2}{-5}\right)^{-(-3)} $$

Simplify the exponent

In this case, the exponent is negative. When we have a negative exponent, we can rewrite the expression as the reciprocal of the base raised to the positive exponent: $$ \left(\frac{1}{\frac{2}{-5}}\right)^{3} $$

Simplify the base

Now, we will rewrite the fraction in the base. To do this, flip the numerator and denominator inside the parentheses: $$ \left(\frac{-5}{2}\right)^{3} $$

Evaluate the expression

Raise the simplified base to the power of 3, which means multiplying the fraction by itself three times: $$ \left(\frac{-5}{2}\right)^{3} = \frac{-5}{2} \cdot \frac{-5}{2} \cdot \frac{-5}{2} $$ Multiply the numerators and denominators together: $$ \frac{(-5)(-5)(-5)}{(2)(2)(2)} = \frac{-125}{8} $$ The expression evaluates to: $$ \left(\frac{x}{z}\right)^{-y} = \frac{-125}{8} $$

Key concepts

Negative Exponents

In mathematics, exponents are used to denote the number of times a base number is multiplied by itself. When dealing with **negative exponents**, understanding a simple rule helps: a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.
For example, when we see \( a^{-n} \), it means \( \frac{1}{a^n} \).

• This transformation turns the operation of raising a number to a power into a fraction.
• It is a handy method to simplify and manipulate expressions that involve powers.

Consider the expression \( \left(\frac{x}{z}\right)^{-y} \). Applying what we've discussed, this expression simplifies as \( \left(\frac{1}{\left(\frac{x}{z}\right)}\right)^y \). For negative exponents, always remember **reciprocal first** then apply the exponent.

Fraction Simplification

Fraction simplification lays at the foundation of many algebra problems, including those involving exponents or intricate algebraic manipulation. Simplifying a fraction involves rewriting it in the form that is easiest to work with, which often means making the numerator and denominator smaller or rearranging their terms to make them cleaner.
To simplify a fraction, identify any common factors between the numerator and denominator:

• Divide both the numerator and denominator by their greatest common factor (GCF).
• Simplify complex expressions within the fraction by performing operations such as flipping a fraction when taking reciprocals.

In our example, flipping the fraction \( \frac{x}{z} \) means simplifying to \( \frac{z}{x} \), which involves changing the positions of the numerator and denominator. This step plays a crucial role when applying the rules of negative exponents.

Exponentiation

Exponentiation is the process of raising a number, known as the base, to a power, known as the exponent. Simply put, it's repeated multiplication of the base number by itself.

• The process involves taking a number, say \(a\), and multiplying it by itself \(n\) times if the exponent is \(n\).
• For example, \(2^3 = 2 \cdot 2 \cdot 2 = 8\).

In our scenario, when the base \(\left(\frac{-5}{2}\right)\) is raised to the power of 3, it implies multiplying the base fraction by itself 3 times, resulting in \( \frac{(-5)(-5)(-5)}{(2)(2)(2)}\). Exponentiation requires careful arithmetic, especially when dealing with negative numbers, as it influences the sign of the final result.

Substitution in Algebra

Substitution is a fundamental algebraic operation. It involves replacing variables with their respective given values to evaluate or simplify expressions.

• This method turns expressions with variables into something more concrete and numerical.
• Substitution simplifies complex equations and helps in finding specific outputs for given scenarios.

In the exercise, substitution is performed by replacing \(x\), \(y\), and \(z\) with their values: 2, -3, and -5 respectively. By substituting these values into the expression \(\left(\frac{x}{z}\right)^{-y}\), we work with specific numbers. Post substitution, it transforms into \(\left(\frac{2}{-5}\right)^{-(-3)}\). Successful substitution sets the stage for applying further algebraic simplifications and operations.