Question

Write each expression as a product or a quotient. Assume all variables are positive. $$ 10^{4-z} $$

Short answer

Question: Rewrite the expression \(10^{4-z}\) as a product or a quotient, assuming all variables are positive. Answer: \(\frac{10^4}{10^z}\)

Step-by-step solution

Identify the base and exponent

In the given expression, the base is 10 and the exponent is \(4-z\). $$ 10^{4-z} $$

Use the exponent subtraction property

The property of exponent subtraction states that \(a^{m-n} = \frac{a^m}{a^n}\). We can rewrite the given expression using this property: $$ 10^{4-z} = \frac{10^4}{10^z} $$

The final expression

Now we have rewritten the expression as a quotient: $$ 10^{4-z} = \frac{10^4}{10^z} $$

Key concepts

Exponent Subtraction

Exponent subtraction is a useful property in algebra that allows us to simplify expressions with the same base having exponents subtracted from each other. If you are given an expression of the form \(a^{m-n}\), where \(a\) is the base, \(m\) is the initial exponent, and \(n\) is the subtracted exponent, you can express this as a quotient: \(\frac{a^m}{a^n}\). This property helps in simplifying complex expressions by turning them into simpler fractions or products. Here’s how it works:

• The base \(a\) remains the same for both the numerator and the denominator.
• The first exponent, \(m\), becomes the exponent of the numerator \(a^m\).
• The second exponent, \(n\), becomes the exponent of the denominator \(a^n\).

For example, in the expression \(10^{4-z}\), it can be rewritten as \(\frac{10^4}{10^z}\). This is an application of the exponent subtraction property where 4 is the original exponent and \(z\) is subtracted from it. This simplifies the expression considerably.

Base and Exponent

Understanding the concepts of base and exponent is crucial when dealing with exponentiation. The base is the number that gets multiplied by itself, while the exponent tells us how many times the base is used in the multiplication. For example, in the expression \(10^{4-z}\):

• The base is 10, which is the number that is multiplied several times.
• The exponent is \(4-z\), which indicates the power to which the base 10 is raised.

The base does not change as you apply exponent rules; it remains constant. However, the exponent can change through operations such as addition, subtraction, multiplication, or division. Understanding how these two components interact is key to solving problems that involve exponentiation.

Quotient

A quotient is the result of division between two numbers or expressions. It is frequently encountered in algebra when reorganizing complex expressions or when applying exponent subtraction. When you apply exponent subtraction to an expression like \(10^{4-z}\), you turn it into a quotient, for instance, \(\frac{10^4}{10^z}\). This represents the division of \(10^4\) by \(10^z\), and it simplifies the calculation by converting it into a straightforward division problem.Understanding how to write expressions as quotients helps clarify calculations, especially when you perform further arithmetic operations. It reveals the relationship between numbers or expressions, showing how one is a multiple (or part) of another.