Problem 24 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 9^{-1} $$

Step-by-Step Solution

Answer

The value of the expression \(9^{-1}\) is \(1 / 9\)

Step 1: Apply the negative exponent rule

Using the rule of negative exponentiation, the expression \(9^{-1}\) can be restated as \(1 / 9^{1}\)

Step 2: Evaluate the expression

The expression \(1 / 9^{1}\) simplifies to \(1 / 9\)

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power or exponent. The exponent indicates how many times the base is multiplied by itself. For example, in the expression \(9^2\), 9 is the base, and 2 is the exponent, meaning \(9\) is multiplied by itself, resulting in \(9 \times 9 = 81\).
With negative exponents, things get a little more interesting. A negative exponent indicates the reciprocal of the base raised to the positive exponent. That means \(9^{-1}\) becomes \(1 / 9\) (which shows the reciprocal of 9).

• Always remember, a negative exponent flips the position of the base with respect to the numerator and denominator.
• An exponent of -1 specifically reverses the base.
• Overall, the use of exponents, whether positive or negative, is a powerful tool in mathematics for simplifying complex expressions and calculations.

Simplifying expressions

Simplifying expressions involves rewriting them in a simpler or more compact form without changing their value. When we simplify, the goal is to make the expression as straightforward as possible. Consider expressions with exponents, for example.
One common rule for simplification is using the properties of exponents, such as:

• The product of powers rule, which states \(a^m \times a^n = a^{m+n}\).
• The power of a power rule, \((a^m)^n = a^{m \times n}\).
• The negative exponent rule, which says that \(a^{-n} = 1/a^n\). This rule transforms an expression like \(9^{-1}\) into \(1/9^1\), making it easier to understand.

In the example of \(9^{-1}\), simplifying led us directly to the fraction \(1/9\), which is the more understandable and easily interpreted form of the expression.

Reciprocal of a number

The reciprocal of a number is essentially "flipping" it in terms of its placement in a fraction. More precisely, the reciprocal of a non-zero number \(a\) is \(1/a\).
This concept is closely related to negative exponents. A number raised to the power of a negative exponent actually tells us to find its reciprocal.

• The reciprocal moves the number from the numerator to the denominator, or vice versa.
• For positive whole numbers like 9, the reciprocal is \(1/9\).
• Reciprocals are essential in various mathematical operations, such as division, where dividing by a number is equivalent to multiplying by its reciprocal.

So, when you encounter a negative exponent, remember it's essentially asking you to take the reciprocal of the base, making calculations and expressions much simpler.