Question Video: Solving Exponential Equations Using Laws of Exponents Mathematics

Video Transcript

Solve five to the power of 𝑥 multiplied by six to the power of negative 𝑥 is equal to 25 over 36.

In order to answer this question, we need to recall some of our laws of exponents. 𝑎 to the power of negative 𝑥 is equal to one over 𝑎 to the power of 𝑥. This means that we can rewrite the left-hand side of our equation as five to the power of 𝑥 multiplied by one over six to the power of 𝑥. This in turn simplifies to five to the power of 𝑥 over six to the power of 𝑥. This is equal to 25 over 36.

Another of our laws of exponents states that 𝑎 to the power of 𝑥 divided by 𝑏 to the power of 𝑥 is equal to the fraction 𝑎 over 𝑏 all raised to the power of 𝑥. This means that the left-hand side of our equation can be rewritten as five-sixths to the power of 𝑥. As five squared is equal to 25 and six squared is equal to 36, the right-hand side can be rewritten as five-sixths squared.

Our base numbers are now the same, which means the exponents must be equal. The solution to the equation five to the power of 𝑥 multiplied by six to the power of negative 𝑥 equals 25 over 36 is 𝑥 equals two.