Video Transcript
Simplify the expression 𝑥 to the eighth power divided by 𝑦 to the negative fourth power all to the power of one-half.
In order to answer this question, we will need to use some of our laws of exponents or indices. Firstly, we recall that when raising a fraction to any power, we can raise the numerator and denominator to the power separately. This means that our expression can be rewritten as 𝑥 to the eighth power to the power of a half over 𝑦 to the negative fourth power to the power of one-half.
We also recall that when a term raised to a power is raised to another power, we can multiply the exponents. 𝑎 to the power of 𝑥 all raised to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 multiplied by 𝑦. Eight multiplied by one-half is equal to four. Therefore, the numerator simplifies to 𝑥 to the fourth power. Negative four multiplied by one-half is equal to negative two. Therefore, the denominator simplifies to 𝑦 to the power of negative two.
Finally, we can use the fact that one over 𝑎 to the power of 𝑥 is the same as 𝑎 to the power of negative 𝑥. This means that one over 𝑦 to the power of negative two is the same as 𝑦 to the power of negative negative two. This in turn simplifies to 𝑦 to the power of two or 𝑦 squared.
The expression 𝑥 to the eighth power over 𝑦 to the power of negative four all raised to the power of one-half in its simplified form is 𝑥 to the fourth power multiplied by 𝑦 squared.
