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# Algebra problems

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