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simplify each of the following in indices
a. s³÷t⁴ x (s-³ t-²)³
b. (4a^3 b^-1 c^-2)^1/2 ÷ [64(a^6 b^4 c^2)^-1/2]
c. 3√8a^3 b^6 ÷ √1/25 a⁴ b^7 * (16√a⁴b^6)^-1/2
Reiny
answered
I will do one of them, you show me what you have done so far for the others.
(4a^3 b^-1 c^-2)^1/2 ÷ [64(a^6 b^4 c^2)^-1/2]
= (2 a^(3/2)(b^(-1/2)(c-1) * (1/64)(a^3 b^2 c^-2)
= (1/32)(a^(9/2)(b^3/2)(c^-3)
= 1/32 √(a^9 b^3)
Explain Bot
answered
Sure! Let's simplify each expression one by one.
a. s³÷t⁴ x (s-³ t-²)³
To simplify this expression, we first need to apply the exponent rules. We'll start by simplifying the exponents inside the brackets.
(s-³ t-²)³ = s³(-3) t⁶(-2) = s⁻⁹ t⁻¹²
Now, we can simplify the whole expression:
s³÷t⁴ x (s-³ t-²)³ = s³÷t⁴ x s⁻⁹ t⁻¹²
To multiply two terms with the same base, we add the exponents:
s³ x s⁻⁹ = s³ + (-9) = s⁻⁶
t⁴ x t⁻¹² = t⁴ + (-12) = t⁻⁸
Therefore, the simplified expression is:
s⁻⁶ t⁻⁸
b. (4a^3 b^-1 c^-2)^1/2 ÷ [64(a^6 b^4 c^2)^-1/2]
To simplify this expression, we can start by simplifying the exponents inside the brackets and applying the power of a power rule:
(4a³ b⁻¹ c⁻²)¹/² = 4¹/² a³(1/²) b⁻¹(1/²) c⁻²(1/²) = 2a^(3/2) b^(-1/2) c^(-1)
[64(a⁶ b⁴ c²)⁻¹/²] = 64^(-1/²) a⁶(-1/²) b⁴(-1/²) c²(-1/²) = 1/(8a^(3/2) b^(1/2) c)
Now, we can simplify the expression using division of exponents:
(2a^(3/2) b^(-1/2) c^(-1)) ÷ (1/(8a^(3/2) b^(1/2) c))
When dividing, we subtract exponents:
2a^(3/2) b^(-1/2) c^(-1) x 8a^(3/2) b^(1/2) c
Multiplying, we add exponents:
16a^(3/2 + 3/2) b^(-1/2 + 1/2) c^(-1)
Simplifying exponents:
16a³ b⁰ c⁻¹ = 16a³c⁻¹
Therefore, the simplified expression is:
16a³c⁻¹
c. 3√(8a³ b⁶) ÷ √(1/25 a⁴ b⁷) x (16√(a⁴b⁶))^(-1/2)
To simplify this expression, let's simplify each radical individually.
3√(8a³ b⁶) = 2a√(a³ b⁶)
√(1/25 a⁴ b⁷) = 1/5 a² b^(7/2)
16√(a⁴b⁶) = 4√(a⁴b⁶)
Now, let's simplify the whole expression:
2a√(a³ b⁶) ÷ (1/5 a² b^(7/2)) x (4√(a⁴b⁶))^(-1/2)
Since we have fractional exponents, we need to apply the power of a product rule and power of a quotient rule.
Applying the power of a product rule:
2a√(a³ b⁶) x 4⁻¹/² (a⁴b⁶)⁻¹/²
Simplifying radicals:
2a (a³)^(1/2) (b⁶)^(1/2) x 4⁻¹/² (a⁴b⁶)⁻¹/²
2a a^(3/2) b^(3) x 4⁻¹/² a^(-2) b^(-3)
Multiplying and simplifying exponents:
2(4⁻¹/²)(a^(3/2 + (-2)))(b^(3 + (-3)))
Simplifying:
2(1/2)(a^(1/2))(1)
Finally, the simplified expression is:
a^(1/2)