Question

Simplify each expression. \(4^{-2} \cdot 4^{3}\)

Short answer

4

Step-by-step solution

Understand the Properties of Exponents

When multiplying expressions with the same base, you add the exponents. The general rule is: \[a^m \times a^n = a^{m+n}\]

Apply the Rule

Using the rule, apply it to the given expression. Here, the base is 4, so: \[4^{-2} \times 4^3 = 4^{-2+3}\]

Simplify the Exponent

Now, simplify the exponent: \[-2 + 3 = 1\] So: \[4^{-2} \times 4^3 = 4^1\]

Write the Final Answer

Any number raised to the power of 1 is the number itself. Therefore: \[4^1 = 4\]

Key concepts

simplifying exponents

Let's start with understanding how to simplify exponents. Exponents are a shorthand notation to express repeated multiplication of a number by itself. For example, instead of writing 2 \times 2 \times 2, you can write 2^3.
Simplifying exponents often involves using the rules of exponents to make expressions easier to work with. Whether adding or subtracting exponents, changing negative exponents to fractions, or breaking down complex exponent expressions, the key is recognizing patterns and rules.
For instance, if you have an expression like \(4^{-2} \times 4^{3}\), you can apply these rules to simplify it.
By combining the exponents because of the same base, you simplify the equation step by step.

multiplication of exponents

Multiplying exponents can be a straightforward process if you understand the basic principles. When you multiply exponential expressions that share the same base, you add their exponents together.
The rule for multiplication is: \ (a^m \times a^n = a^{m+n}) \.
This rule only applies if the bases are the same. Similarly, in our expression of 4^{-2} times 4^3, since both terms have the base 4, we use the rule:
\[4^{-2} \times 4^3 = 4^{-2+3}\frac\]
We combined the exponents by adding -2 and 3. This gives us the simplified exponent 1, so it equals 4^1.

base and exponent

Understanding the components of an exponent is crucial for handling these expressions correctly. A base is the main number being multiplied, while the exponent indicates how many times the base is used in the multiplication.
For example, in 4^3, 4 is the base, and 3 is the exponent. The expression means 4 multiplied by itself three times.
In the original exercise \(4^{-2} \times 4^{3}\), both bases are the same, which simplifies the process. When you correctly apply the multiplication rule for exponents, you get:
\(4^{-2+3} = 4^1\). And since any number raised to the power of 1 is itself, the final answer is simply 4.