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Simplify the expression two 𝑥 to the power of one-fifth 𝑦 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 𝑥 to the power of one-fifth 𝑦 to the power of one-seventh multiplied by two 𝑥 to the power of one-fifth 𝑦 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 𝑎 to the power of 𝑥 multiplied by 𝑎 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 plus 𝑦. 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 𝑥 to the power of one-fifth by 𝑥 to the power of one-fifth. As one-fifth plus one-fifth is equal to two-fifths, this is equal to 𝑥 to the power of two-fifths. We repeat this with the 𝑦-parts of our expression. One-seventh plus one-seventh is equal to two-sevenths, giving us 𝑦 to the power of two-sevenths. The simplified version of our expression is therefore equal to four 𝑥 to the power of two-fifths 𝑦 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 𝑥 to the power of one-fifth squared multiplied by 𝑦 to the power of one-seventh squared. We can then use the fact that 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. 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 𝑥 to the power of two-fifths. One-seventh multiplied by two is equal to two-sevenths, giving us 𝑦 to the power of two-sevenths. This confirms that the expression two 𝑥 to the power of one-fifth 𝑦 to the power of one-seventh squared is equal to four 𝑥 to the power of two-fifths 𝑦 to the power of two-sevenths.