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The term that does not contain an irrational expression in the expansion of (3¹/⁵ ⋅2¹/⁷)²⁴ is____

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Final answer:

The term in the expression (31/5 · 21/7)24 that does not contain an irrational expression is 648, because the exponents become whole numbers when multiplying by the power of 24.

Step-by-step explanation:

The term in the expansion of (31/5 · 21/7)24 that does not contain an irrational expression can be found using exponent properties. When we raise a product to a power, we raise each factor to that power separately and then multiply the results. Hence, the expression becomes (31/5)24 · (21/7)24. We then multiply the exponents and get 324/5 · 224/7. To find the term without an irrational expression, we need to have integral exponents.

In the expansion, the term with the base 3 and the exponent 1/5 will be irrational, because raising a number to a fractional power can lead to an irrational result. The term with the base 2 and the exponent 1/7 will also be irrational.

However, the term with the base 2 and the exponent 72 will be rational, as the resulting exponent is a whole number. Hence, the term that does not contain an irrational expression in the expansion is 2<sup>72</sup>.

Since 24 is divisible by both 5 and 7, the exponents 24/5 and 24/7 are whole numbers, meaning they won't result in irrational numbers. Thus, the term is simply 34 · 23, which equals 81 · 8 = 648. Therefore, the term that does not contain an irrational expression is 648.

User Pierre Clocher
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