Question

78. Use properties of exponents to rewrite each expression as either an integer or as a quotient of integers p/q to show the expression is a rational number.
a.4 √2. 4 √8

Answer 1

Answer:

4√2 × 4√8 = 64, an integer.

Step-by-step explanation:

a) 4√2 × 4√8

First let's remember that a square root can be rewritten as a 1/2 exponent,

Therefore the expression we have can be rewritten as

$4(2^{1/2} )4(8^{1/2})$

We are going to write all numbers in their prime factor expressions:

*Note: Remember that when we multiply two numbers with the same base but different exponents we sum up the exponents*

*Also, when we elevate one exponent to another exponent, we multiply the exponents*

$4(2^{1/2} )4(8^{1/2})\\=2^2 (2^{1/2})2^2 (2^{3})^{1/2} \\ =2^2 (2^{1/2})2^2 (2^{3/2})\\=2^22^2(2^{1/2}2^{3/2})\\=16(2^{4/2})\\=16(2^2)\\=16(4)\\=64$

Therefore, the expression 4√2 × 4√8 = 64 and therefore is an integer