Question

$$ \text { If } x^{a}=\frac{y}{z}, y=x^{b}, \text { and } z=x^{c}, \text { what is } a ? $$

Short answer

#Answer# The value of the exponent 'a' in terms of 'b' and 'c' is: a = b - c.

Step-by-step solution

Write down the given equations

We are given the following equations: 1) \(x^a = \frac{y}{z}\) 2) \(y = x^b\) 3) \(z = x^c\)

Substitute the expressions of y and z in the first equation

Replace y with its expression \(x^b\) and z with its expression \(x^c\) in the first equation: \(x^a = \frac{x^b}{x^c}\)

Simplify the equation using the properties of exponents

Use the quotient rule for exponents which states that \(x^{m-n} = \frac{x^m}{x^n}\): \(x^a = x^{b-c}\)

Equate the exponents to find the value of 'a'

Since the bases are equal (they are both x), their exponents must also be equal: \(a = b-c\) Now we have the value of 'a' in terms of 'b' and 'c'.

Key concepts

Exponents

Exponents are mathematical symbols used to represent repeated multiplication of a number by itself. When you see a number, like 3, raised to the power of another number, say 4, written as \(3^4\), it means you multiply 3 by itself four times. Therefore, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
Exponents are very important because they make it easier to write and work with large numbers. For example, instead of writing 10000, you can write \(10^4\), which is more concise and easier to manage in equations.
Exponents have several rules or properties that help simplify expressions. Some of these include the product rule \(x^m \times x^n = x^{m+n}\), the power rule \((x^m)^n = x^{mn}\), and the quotient rule \(\frac{x^m}{x^n} = x^{m-n}\). These rules are essential for solving problems that involve exponential terms.

Substitution Method

The substitution method is a technique used in algebra to simplify and solve equations. It involves replacing variables with given or known quantities to transform complex expressions into simpler forms.
In the exercise, the substitution method is applied to equations 2 and 3. They express y and z in terms of x as \(y = x^b\) and \(z = x^c\). Substituting these into the first equation, \(x^a = \frac{y}{z}\), results in equations that are entirely in terms of the base x: \(x^a = \frac{x^b}{x^c}\).
By using substitution, we reduce the problem into one variable, which makes the next steps easier because we can apply mathematical properties, such as those of exponents, directly to find a solution.

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions and solving equations involving exponential terms. These properties provide rules on how to manipulate exponents when they are part of larger mathematical expressions.
In our original exercise, we used the quotient rule of exponents: \(\frac{x^m}{x^n} = x^{m-n}\). This property lets us simplify divisions of similar bases with exponents. We applied this by transforming the expression \(\frac{x^b}{x^c}\) into \(x^{b-c}\).
Another key property is when two expressions with the same base are equal, their exponents must also be equal. This principle allowed us to equate \(x^a\) to \(x^{b-c}\) and hence find that \(a = b - c\). Understanding these properties of exponents enables you to handle complex algebraic expressions more effectively.