Enter the correct answer in the box. simplify the expression x5 • x7.
The Correct answer and Explanation is :
To simplify the expression ( x^5 \cdot x^7 ), you apply the product of powers rule from the laws of exponents. This rule states that when you multiply two expressions with the same base, you add their exponents. In this case, the base is ( x ), and the exponents are 5 and 7.
Step-by-Step Simplification:
- Write the original expression: ( x^5 \cdot x^7 ).
- According to the product of powers rule, add the exponents:
[
x^5 \cdot x^7 = x^{5+7}.
] - Simplify the exponent:
[
5 + 7 = 12.
] - The simplified expression is ( x^{12} ).
Thus, the simplified form of ( x^5 \cdot x^7 ) is ( x^{12} ).
Explanation:
The product of powers rule works because exponents represent repeated multiplication. For example, ( x^5 ) means multiplying ( x ) by itself 5 times, and ( x^7 ) means multiplying ( x ) by itself 7 times. When you multiply ( x^5 ) by ( x^7 ), you are combining all the ( x )’s together, which results in multiplying ( x ) by itself 12 times in total.
This rule applies as long as the bases (in this case, ( x )) are the same. You cannot add exponents if the bases are different, like in the expression ( x^5 \cdot y^7 ); they must remain separate unless the bases are the same.
Exponent rules like these are crucial in algebra because they allow you to simplify complex expressions, making equations easier to solve or understand. Simplifying such expressions also makes it easier to evaluate the expression for different values of ( x ).