Question

Simplify and express each of the following in exponential form:

(i) (23 × 33 × 4) / (3 × 32) (ii) ((52 )3 × 54) ÷ 57 (iii) 254 ÷ 53
(iv) (3 × 72 ×118) / (21×113) (v) 37 / (34 × 33) (vi) 20 + 30 + 40
(vii) 20 × 30 × 40 (viii) (30 + 20 )× 50 (ix) (28 × a5) / (43 × a3)
(x) (a5 / a3) × a8 (xi) (45 × a8b3) / (45 × a5b2) (xii) (23 × 2)2

Answer 1

The student's question involves simplifying expressions with exponential forms using the laws of exponents, including the multiplication rule, the power rule, and the division rule. Each expression is simplified by applying the relevant rules step-by-step.

Step-by-step explanation:

The student is asking about simplifying expressions with exponential forms. Below are the step-by-step explanations for the given expressions:

1. (23 × 33 × 4) / (3 × 32): First factor out common bases and apply the laws of exponents to simplify the expression.
2. ((52)3 × 54) ÷ 57: Apply the cubing of exponentials and the division rule for exponents to simplify.
3. 254 ÷ 53: Solve by reducing the exponents where possible.
4. (3 × 72 × 118) / (21 × 113): Apply the division rule for exponents and reduce the expression.
5. 37 / (34 × 33): Simplify by subtracting exponents during division.
6. 20 + 30 + 40: Remember that any number raised to the exponent zero equals one.
7. 20 × 30 × 40: Apply the same rule for exponent zero.
8. (30 + 20) × 50: Use the rule for exponents and simplify.
9. (28 × a5) / (43 × a3): Factor the numerical base and simplify the algebraic exponents.
10. (a5 / a3) × a8: Apply the division rule for exponents and multiply.
11. (45 × a8b3) / (45 × a5b2): Simplify by canceling out the common bases and reducing the exponents of the variables.
12. (23 × 2)2: Combine the bases and apply the power of a product rule.

The concepts related to laws of exponents and their application to simplify expressions play a crucial role in solving these problems.