[tex]\frac{12}{x^7} \times \frac{7}{x^{12}}[/tex]
Answer (2)
Multiply the numerators: 12 × 7 = 84 .
Multiply the denominators using the exponent rule: x 7 × x 12 = x 7 + 12 = x 19 .
Combine the results to form the simplified expression.
The simplified expression is: x 19 84 .
Explanation
Understanding the Problem We are given the expression x 7 12 × x 12 7 . Our goal is to simplify this expression by multiplying the two fractions.
Multiplying Numerators To multiply the fractions, we multiply the numerators and the denominators separately. First, we multiply the numerators: 12 × 7 = 84 .
Multiplying Denominators Next, we multiply the denominators: x 7 × x 12 . Using the exponent rule x a × x b = x a + b , we have x 7 × x 12 = x 7 + 12 = x 19 .
Combining Results Now, we combine the results to get the simplified expression: x 19 84 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating the area or volume of geometric shapes. For example, if you have a rectangle with sides of length x 2 2 and x 3 5 , the area would be x 2 2 × x 3 5 = x 5 10 . This skill is also important in physics when dealing with quantities that involve powers, such as calculating force or energy.
The expression x 7 12 × x 12 7 simplifies to x 19 84 by multiplying the numerators and using the exponent rule for the denominators. The final result is x 19 84 .
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