To begin, we apply the product rule to (125a8) 1/3\s(125a8) 1/3 = 1251/3 (a8) (a8) 1/3
The number 125 is then written as a prime to a power (factor 5 out of 125)
1251/3 (a8) (a8)
1/3 = (53) (53)
1/3 (a8) (a8)
1/3
The exponents are then multiplied using the power rule.
The power rule states that (am) n = amn.
3 is written as a fraction with the denominator 1 (53)
1/3 (a8) (a8)
1/3 =5(3/1.1/3) (a8) (a8)
1/3
We then multiply the exponents.
5(3/1*1/3) (a8) (a8)
1/3 = 5(1/1*1/1) (a8) (a8)
1/3
Because 1/1 divided by 1/1 equals 1/1, we rewrite the expression as 51/1 (a8)1/3.
Nevertheless, 1/1= 1, so 51 (a8)
1/3 = 5(a8)1/3 We again apply power rule ((am) n = amn) to multiply the exponent in (a8)1/3
8(1/3) = 8/3
51 (a8)1/3 = 5(a8)1/3
We again apply power rule ((am) n = amn) to multiply the exponent in (a8)1/3
8(1/3) = 8/3
51 (a8)1/3 =5 a8/3
Therefore, (125a8)1/3 = 5 a8/3
Problem 2
√12+√24
We separate and simply each part of the question before summing them i.e. simple √12 and √24 separately
First, we simplify√12
We begin by factoring 4 out of 12
12= √ (4.3)
We then express √ (4.3) in separate roots
√ (4.3) = √ 4 * √ 3
We then express √ 4 as a prime to a power (factor 2 out of 4)
4 = 22
We then rewrite the expression as:
√ 4 * √ 3 = √ 22* √ 3
The next step is to convert √ 22 to a fractional exponent
Since the nth root of a square root is 2, then
√ 22 = (22)1/2
√ 22* √ 3 = (22)1/2 * √ 3
We then apply the power rule, (am) n = amn to (22)1/2
(22)1/2 = 2(2*1/2) = 21/1. However, 1/1 =1
Therefore, (22)1/2 = 2
The radicand 3 in √ 3 cannot be simplified further because it cannot be expressed as a prime to a power
Therefore, we write the expression as follows
√ 12= 2√ 3
Secondly we simplify √24
We begin by factoring 4 out of 24
24= √ (4.6)
We then express √ (4.6) separately
√ (4.3) = √ 4 * √ 6
We then express √ 4 as a prime to a power (factor 2 out of 4)
4 = 22
We then rewrite the expression as:
√ 4 * √ 6 = √ 22* √ 6
The next step is to convert √ 22 to a fractional exponent
Since the nth root of a square root is 2, then
√ 22 = (22)1/2
√ 22* √ 6 = (22)1/2 * √ 6
We then apply the power rule, (am) n = amn to (22)1/2
(22)1/2 = 2(2*1/2) = 21/1. However, 1/1 =1
Therefore, (22)1/2 = 2
The radicand 6 in √6 cannot be simplified further because it cannot be expressed as a prime to a power
Therefore, √ 24= 2√ 6
We rewrite the original expression as √12+√24 = 2√3+2√6
Since we cannot simplify the expression 2√3+2√6 further because it only contain unlike surds, then we write the final expression as follows
√12+√24 = 2√3+2√6
