Problem 17 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form $$8\left(\frac{1}{4}\right)^{-1}$$

Step-by-Step Solution

Answer

The value of the given expression is 32. The equivalent fraction expression in simplest form is \(32/1\) or just 32.

Step 1: Rewrite the expression using exponent rule for negative number

The given exponential expression is \(8( \frac{1}{4})^{-1}\), using the rule of negative exponents, this expression can be rewritten as an equivalent one: \(8 \cdot 4^1\). The -1 exponent inverts the fraction, changing denominator to numerator.

Step 2: Simplify the expression

Next, we will simplify the expression. Since 4^1 is just 4, the expression becomes \(8 \cdot 4 = 32\).

Key Concepts

Negative Exponents

Understanding negative exponents is crucial to mastering algebra and simplifying exponential expressions. A negative exponent indicates that the base (the number being raised to a power) should be taken as the reciprocal, with the exponent becoming positive. For example, if we have an expression like \( a^{-n} \), it implies \( \frac{1}{a^n} \), inverting the base and making the exponent positive.

Following this rule, if you encounter an expression such as \( \frac{1}{4}^{-1} \) in a problem like \( 8\left(\frac{1}{4}\right)^{-1} \), you would convert \( \frac{1}{4} \) to its reciprocal, which is just 4, and the negative exponent to a positive exponent. The resulting expression, \( 8 \cdot 4^1 \), can then be further simplified. This technique allows for simplification without changing the value of the original expression.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra which helps to make complex problems more manageable. To simplify an expression, combine like terms, use the distributive property, and apply exponent rules effectively.

In our example expression \( 8 \cdot 4^1 \), simplification is straightforward since \( 4^1 \) is simply 4. Once we rewrite the expression as \( 8 \cdot 4 \) and carry out the multiplication, the simplified result is 32. Remember that simplification is not just about calculations; it also involves organizing an expression into its most concise, easily understood form. This often means eliminating unnecessary terms or factors and reducing fractions to their simplest form.

Exponent Rules

The rules for exponents allow us to perform operations on expressions with powers in a consistent and reliable way. Some of these rules include:

• The Product Rule: \( a^m \cdot a^n = a^{m+n} \) - When multiplying expressions with the same base, add the exponents.
• The Power Rule: \( (a^m)^n = a^{m \cdot n} \) - When an exponential expression is raised to a power, multiply the exponents.
• The Zero Exponent Rule: \( a^0 = 1 \) - Any base raised to the power of zero is one.
• The Negative Exponent Rule, as discussed above, whereby \( a^{-n} = \frac{1}{a^n} \) - Any number with a negative exponent becomes a fraction with a positive exponent.

Applying Exponent Rules

In the exercise \( 8\left(\frac{1}{4}\right)^{-1} \) the negative exponent rule is aptly applied. Simplifying exponential expressions often involves multiple rules, and understanding how to apply them in combination is key to solving more complex problems efficiently and correctly.