Let's analyze the mathematical expressions visible in the image.

The instructions are in Indonesian and say:

Latihan 7

• Kerjakan soal-soal berikut dengan benar! (Solve the following problems correctly!)

1. Sederhanakan bentuk-bentuk berikut! (Simplify the following expressions!)

The problems are:

a. $\left(a^{\frac{2}{3}}\right)^2 \times a^{\frac{1}{2}}$

b. $\left(\frac{a^{\frac{2}{3}} b^{-3}}{a^{-3} b^{-2}}\right)^{\frac{3}{4}}$

Let’s start by simplifying each of these.

Solution for (a):

Expression: $\left(a^{\frac{2}{3}}\right)^2 \times a^{\frac{1}{2}}$

1. First, apply the power of a power rule: $\left(a^{\frac{2}{3}}\right)^2 = a^{\frac{4}{3}}$

2. Multiply the exponents of $a$: $a^{\frac{4}{3}} \times a^{\frac{1}{2}} = a^{\frac{4}{3} + \frac{1}{2}}$

3. Find the common denominator and add the fractions: $\frac{4}{3} + \frac{1}{2} = \frac{8}{6} + \frac{3}{6} = \frac{11}{6}$

Final simplified form: $a^{\frac{11}{6}}$

Solution for (b):

Expression: $\left(\frac{a^{\frac{2}{3}} b^{-3}}{a^{-3} b^{-2}}\right)^{\frac{3}{4}}$

1. Simplify inside the parentheses: $\frac{a^{\frac{2}{3}}}{a^{-3}} = a^{\frac{2}{3} - (-3)} = a^{\frac{2}{3} + 3} = a^{\frac{2}{3} + \frac{9}{3}} = a^{\frac{11}{3}}$ $\frac{b^{-3}}{b^{-2}} = b^{-3 - (-2)} = b^{-3 + 2} = b^{-1}$

   Now the expression becomes: $\left(a^{\frac{11}{3}} b^{-1}\right)^{\frac{3}{4}}$

2. Apply the exponent to both $a$ and $b$: $a^{\frac{11}{3} \times \frac{3}{4}} = a^{\frac{11}{4}}, \quad b^{-1 \times \frac{3}{4}} = b^{-\frac{3}{4}}$

Final simplified form: $a^{\frac{11}{4}} b^{-\frac{3}{4}}$

Would you like more details on these steps or need further explanations?

Here are five related questions to expand this topic:

1. How do you handle negative exponents in a fraction?
2. How can you simplify expressions with multiple variables raised to different powers?
3. What are some common mistakes when adding fractions with different denominators?
4. How do you apply exponent rules when there are products and quotients combined?
5. How would the rules change if the base was a fraction instead of a variable?

Tip: Always simplify exponents step by step, starting with any powers inside parentheses before multiplying or dividing.