Explanation:
Sure, let's simplify and evaluate these expressions using the laws of exponents:
a) (3x⁴)⁹ ÷ (3⁵x⁶)³
First, apply the power of a power rule: (aⁿ)ⁿ = a^(n * n)
(3x⁴)⁹ = 3^(9 * 4) * x^(4 * 9)
Now, simplify the denominator using the same rule: (3⁵x⁶)³ = 3^(5 * 3) * x^(6 * 3)
Now, the expression becomes:
(3^(36) * x^(36)) ÷ (3^(15) * x^(18))
Next, apply the quotient rule for exponents: aⁿ / a^m = a^(n - m)
So, (3^(36) * x^(36)) ÷ (3^(15) * x^(18)) = 3^(36 - 15) * x^(36 - 18)
Simplify further:
3^21 * x^18
Now, you can evaluate this expression if you have a specific value for x.
b) [(6³a⁵)/(5³b⁹)]⁷×[(5¹¹b²)/(6⁶a³)]³
First, let's simplify each part separately using the power of a quotient rule: (a/b)ⁿ = aⁿ / bⁿ
For the first part:
(6³a⁵)/(5³b⁹) becomes 6^(3 * 7) * a^(5 * 7) / (5^(3 * 7) * b^(9 * 7))
So, it becomes:
6^21 * a^35 / 5^21 * b^63
Now, for the second part:
(5¹¹b²)/(6⁶a³) becomes 5^(11 * 3) * b^(2 * 3) / (6^(6 * 3) * a^(3 * 3))
This simplifies to:
5^33 * b^6 / 6^18 * a^9
Now, we have:
(6^21 * a^35 / 5^21 * b^63)⁷ × (5^33 * b^6 / 6^18 * a^9)³
Apply the power of a product rule: (ab)ⁿ = aⁿ * bⁿ
(6^21 * a^35 / 5^21 * b^63)⁷ × (5^33 * b^6 / 6^18 * a^9)³ becomes:
(6^(21 * 7) * a^(35 * 7) / 5^(21 * 7) * b^(63 * 7)) × (5^(33 * 3) * b^(6 * 3) / 6^(18 * 3) * a^(9 * 3))
Now, calculate each part separately and then multiply:
6^147 * a^245 / 5^147 * b^441 × 5^99 * b^18 / 6^54 * a^27
Apply the power of a quotient rule to each part:
(6^147 / 6^54) * (a^245 / a^27) * (5^99 / 5^147) * (b^18 / b^441)
Now, simplify each part:
6^(147 - 54) * a^(245 - 27) * 5^(99 - 147) * b^(18 - 441)
6^93 * a^218 * 5^(-48) * b^(-423)
Now, you can evaluate this expression if you have specific values for a and b.