Video Transcript
Simplify the expression two π₯ to the power of one-fifth π¦ to the power of one-seventh all squared.
There are a couple of ways of approaching this problem. One way would be to recall that when we square a term, we multiply it by itself. This means that we could rewrite the expression as two π₯ to the power of one-fifth π¦ to the power of one-seventh multiplied by two π₯ to the power of one-fifth π¦ to the power of one-seventh. We could then use one of our laws of exponents or indices to simplify this expression. We know that π to the power of π₯ multiplied by π to the power of π¦ is equal to π to the power of π₯ plus π¦. When multiplying terms with the same base, we simply add the exponents. Two multiplied by two is equal to four.
Next, we need to multiply π₯ to the power of one-fifth by π₯ to the power of one-fifth. As one-fifth plus one-fifth is equal to two-fifths, this is equal to π₯ to the power of two-fifths. We repeat this with the π¦-parts of our expression. One-seventh plus one-seventh is equal to two-sevenths, giving us π¦ to the power of two-sevenths. The simplified version of our expression is therefore equal to four π₯ to the power of two-fifths π¦ to the power of two-sevenths.
An alternative method would be to rewrite the expression by squaring each of the individual parts. This would give us two squared multiplied by π₯ to the power of one-fifth squared multiplied by π¦ to the power of one-seventh squared. We can then use the fact that π to the power of π₯ all raised to the power of π¦ is equal to π to the power of π₯ multiplied by π¦. We begin by calculating two squared which is equal to four.
Next, we need to multiply the exponents one-fifth and two. This is equal to two-fifths. So we have π₯ to the power of two-fifths. One-seventh multiplied by two is equal to two-sevenths, giving us π¦ to the power of two-sevenths. This confirms that the expression two π₯ to the power of one-fifth π¦ to the power of one-seventh squared is equal to four π₯ to the power of two-fifths π¦ to the power of two-sevenths.