Write six to the power of two-thirds multiplied by six to the power of six-eighths in radical form.
We firstly need to consider one of our laws of indices: 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. When we are multiplying two numbers with the same base, we can add the exponents or powers. In this case, we need to add two-thirds and six-eighths. The lowest common multiple of three and eight is 24. Three multiplied by eight is 24. And eight multiplied by three is 24. As we’ve multiplied the denominator of the first fraction by eight, we need to multiply the numerator by eight. Two multiplied by eight is 16. We multiply the denominator of the second fraction by three. Multiplying the numerator by three gives us 18, as six multiplied by three is 18.
16 24ths plus 18 24ths is equal to 34 24ths or 34 over 24. We can simplify this fraction by dividing the top, the numerator, and the bottom, the denominator, by two. This gives us 17 12ths.
This means that six to the power of two-thirds multiplied by six to the power of six-eighths is equal to six to the power of 17 12ths.
As 𝑥 to the power a half is equal to the square root of 𝑥, 𝑥 to the power a third is equal to the cube root of 𝑥, this means that 𝑥 to the power of one over 𝑛 is equal to the 𝑛th root of 𝑥. This means that we can rewrite six to the power of 17 12ths as the 12th root of six to the power of 17.
So six to the power of two-thirds multiplied by six to the power of six-eighths, in radical form, is the 12th root of six to the power of 17.