Question

Write an expression in factored form that is equivalent to the expression 1/8x+16

Answer 1

Step-by-step explanation: In order to factor an integer (the 8 in the fraction 1/8) we need to repeatedly divide it by the ascending sequence of primes (2, 3, 5)

The number that each prime divides the original integer becomes it's exponent in the final result.

In the example: $\frac{1}{8} x+16$

Prime number 2 to the power of 3 equals 8.

$\frac{1}{2^3} x+16$

Now, we need to perform a multiplication. The following rule is applied: $\frac{A}{B} C=\frac{AC}{B}$

In the example: $\frac{1}{2^3} x+16$

The new factors in the numerator are: 1x.

Notice that all non-fractions are placed in the numerator.

The factors in the new denominator are: $2^3$

$\frac{x}{2^3} +16$

Now, we need to add the fractions. We use this rule: $\frac{A}{B} +\frac{C}{D} =\frac{\frac{LCD}{B}A+\frac{LCD}{D}C }{LCD}$

The example involved 2 terms. $\frac{x}{2^3} +16$

Note that 1 non-fractional term are treated as fractions with a denominator equal to 1.

The LCD is: $2^3$

$\frac{x}{2^3} +16$changes to$\frac{x+(2^3)16}{2^3}$

We need to evaluate a power by multiplying the base by itself as many times as the exponent indicates.

In the example: $\frac{x+(2^3)16}{2^3}$  (the one in parenthesis)

Base 2 will be multiplied by itself 3 times.

2 x 2 x 2 = 8 ($2^3$ = 8)

$\frac{x+8\times16}{2^3}$ (Notice that $2^3$ is now 8 in the fraction.)

Lastly, we multiply the 8 by 16 in the fraction, $\frac{x+8\times16}{2^3}$

8 x 16 = 128

$\frac{x+128}{2^3}$ (Notice that 8 x 16 is now 128 in the fraction)

Therefore, your answer is: $\frac{x+128}{2^3}$