Sure! Let's go through the simplification and expansion step by step.
### (1) Simplify these:
a) [tex]\( \frac{3}{5}x - \frac{2}{3}x \)[/tex]
To simplify this expression, we need to combine the terms by finding a common denominator.
1. Rewrite [tex]\( \frac{3}{5} \)[/tex] and [tex]\( \frac{2}{3} \)[/tex] with a common denominator of 15:
[tex]\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\][/tex]
[tex]\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\][/tex]
2. Subtract the fractions:
[tex]\[
\frac{3}{5}x - \frac{2}{3}x = \frac{9}{15}x - \frac{10}{15}x
\][/tex]
[tex]\[
= \frac{9x - 10x}{15}
\][/tex]
[tex]\[
= \frac{-x}{15} = -\frac{1}{15}x
\][/tex]
So, the simplified form is:
[tex]\[ -\frac{1}{15}x \][/tex]
Which simplifies numerically to:
[tex]\[ -0.0666666666666667x \][/tex]
b) [tex]\(\frac{x^2 - 2x - 48}{x^2 + 4x + 48}\)[/tex]
To simplify this rational expression, we look for common factors in the numerator and denominator.
1. Given the numerator:
[tex]\[
x^2 - 2x - 48
\][/tex]
2. Given the denominator:
[tex]\[
x^2 + 4x + 48
\][/tex]
The numerator and the denominator do not have common factors that further simplify in an evident way with basic arithmetic.
Thus, the simplified form remains:
[tex]\[ \frac{x^2 - 2x - 48}{x^2 + 4x + 48} \][/tex]
### (2) Simplify and Expand these:
c) [tex]\(\frac{\log 8}{\log 3^2}\)[/tex]
1. Use the properties of logarithms to simplify:
[tex]\[
\log 3^2 = \log 9
\][/tex]
[tex]\[
\frac{\log 8}{\log 9}
\][/tex]
### d) [tex]\(\frac{3 \log 2 + \log 4}{\log 8}\)[/tex]
1. Split and simplify each term using properties of logarithms:
[tex]\[
\log 4 = \log(2^2) = 2 \log 2
\][/tex]
2. Combine the terms in the numerator:
[tex]\[
3 \log 2 + \log 4 = 3 \log 2 + 2 \log 2 = 5 \log 2
\][/tex]
3. Use properties of logarithms for the denominator:
[tex]\[
\log 8 = \log(2^3) = 3 \log 2
\][/tex]
4. Form the fraction:
[tex]\[
\frac{5 \log 2}{3 \log 2}
\][/tex]
5. Since [tex]\(\log 2\)[/tex] is common in both numerator and denominator, they cancel each other out:
[tex]\[
\frac{5 \log 2}{3 \log 2} = \frac{5}{3}
\][/tex]
So, the simplified form is:
[tex]\[ \frac{5}{3} \][/tex]
