What is the equivalent expression of $x^3 x^4 ?$
A. $\frac{1}{x}$
B. $x^{12}$
C. $x^7$
D. $x$
Answer (1)
Use the rule of exponents: x a x b = x a + b .
Apply the rule to the given expression: x 3 x 4 = x 3 + 4 .
Simplify the exponent: 3 + 4 = 7 .
The equivalent expression is x 7 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to x 3 x 4 . To do this, we need to remember the rules of exponents. When multiplying terms with the same base, we add the exponents. In other words, x a × x b = x a + b .
Applying the Rule of Exponents Applying this rule to our expression, we have x 3 x 4 = x 3 + 4 .
Simplifying the Exponent Now we simply add the exponents: 3 + 4 = 7 . Therefore, x 3 x 4 = x 7 .
Finding the Answer Comparing our result to the given options, we see that the equivalent expression is x 7 , which corresponds to option C.
Examples
Understanding exponents is crucial in many areas, such as calculating compound interest. For example, if you invest $100 at an annual interest rate of 5%, the amount you have after 3 years can be calculated as 100 × ( 1.05 ) 3 . This is an example of exponential growth, where the exponent indicates the number of times the base (1.05) is multiplied by itself. Simplifying expressions with exponents helps in understanding and predicting such growth patterns.
