Question

Divide and write your answer without negative exponents. $$\frac{10^{b-1}}{10^{b-3}}$$

Short answer

The expression \(\frac{10^{b-1}}{10^{b-3}}\) simplifies to \(10^2\) or 100.

Step-by-step solution

Apply the Quotient of Powers Rule

When dividing two exponentials with the same base, subtract the exponent in the denominator from the exponent in the numerator: $$\frac{10^{b-1}}{10^{b-3}} = 10^{(b-1)-(b-3)}.$$

Simplify the Exponent

Subtract the exponents, taking care to distribute the negative sign:$$10^{(b-1)-(b-3)} = 10^{b-1-b+3}.$$ Combine like terms in the exponent:$$10^{b-1-b+3} = 10^{2}.$$

Write the Final Answer

The division of the two exponents results in:$$\frac{10^{b-1}}{10^{b-3}} = 10^2.$$

Key concepts

Exponents

Exponents are a shorthand way in mathematics to represent repeated multiplication of the same number. An exponent consists of a base and a power. The base is the number that is multiplied by itself, and the power, also known as the exponent, tells us how many times the base is to be multiplied. For example, when we write \( 10^3 \), it means we multiply the base, which is 10, by itself three times: \( 10 \times 10 \times 10 \).

Understanding how exponents work is vital because they are used across various domains of mathematics, including algebra, geometry, and calculus. Exponents can simplify how we perform calculations with large numbers and express very large or very small quantities efficiently. Consequently, mastering the rules that govern their operations, like the Quotient of Powers rule, is essential for simplifying mathematical expressions.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its most basic form without changing its value. The process can include combining like terms, factoring, expanding expressions, and applying rules of exponents. The goal when simplifying is to make the expression as clear and concise as possible.

For example, using the Quotient of Powers rule simplifies the division of two powers with the same base. Instead of calculating each power separately, we subtract the exponents. This rule makes it easier to manage expressions with higher powers that could otherwise be time-consuming and prone to calculation errors. In the classroom or tests, simplifying expressions can help in understanding the underlying structure of an equation or expression, making it possible to solve problems more quickly and accurately.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For instance, \( 10^{-2} \) is the same as \( \frac{1}{10^2} \) or \( \frac{1}{100} \). It's crucial to understand that a negative exponent does not make the entire value negative, but rather signifies that the base is on the 'opposite side' of the fraction from where it would be with a positive exponent.

To work with negative exponents, one often needs to rewrite them as positive exponents to simplify expressions fully. This technique is also useful in avoiding negative exponents in final answers, as they can be less intuitive to understand and interpret. The exercise given demonstrates this process by converting a quotient of powers into a single positive exponent, which is a simpler and more digestible form for further mathematical operations or applications.