Simplify & write the answer using positive exponents:
1. (3^3)(3^-6)
2. S^2 t^-1 / (s^4 t^4)^1/2
Simplify then evaluate, express answer to 4 decimal places where nessasssary :
3. [(2/3)^2]^-3
4. [(1-4)^-3 Divide (1/4)^2]^-2
To simplify and write the answers using positive exponents, you need to follow certain rules and principles. Let's go through each question step by step:
1. (3^3)(3^-6):
To simplify this expression, you can use the rule of exponents that states when you multiply two numbers with the same base, you add the exponents. In this case, the base is 3. So, (3^3)(3^-6) is written as 3^(3+(-6)). Simplifying the exponent, 3+(-6) gives you -3. Therefore, the simplified expression is 3^-3.
2. S^2 t^-1 / (s^4 t^4)^1/2:
To simplify this expression, you can use the rule of exponents that states when you raise a power to another power, you multiply the exponents. Additionally, when you divide two numbers with the same base, you subtract the exponents. Furthermore, when you have a square root, it is equivalent to raising the expression to the power of 1/2. Now let's simplify step by step:
First, simplify (s^4 t^4)^1/2. This is equivalent to raising each factor to the power of 1/2: s^(4*1/2) t^(4*1/2) = s^2 t^2 .
Next, substitute this result back into the main expression: S^2 t^-1 / (s^4 t^4)^1/2 becomes S^2 t^-1 / (s^2 t^2).
Now, when dividing, you subtract exponents, so the expression becomes S^2 * t^-1 * s^-2 * t^-2.
Finally, to simplify further, multiply the like terms together: S^2 * s^-2 * t^-1 * t^-2 = S^2 / (s^2 * t^3).
Now you have the simplified expression: S^2 / (s^2 * t^3).
3. [(2/3)^2]^-3:
To simplify this expression, you can use the rule of exponents that states when you raise a fraction to a negative exponent, you can flip the fraction and change the sign of the exponent. Let's simplify the expression step by step:
First, simplify the expression inside the brackets: (2/3)^2 = 4/9.
Next, rewrite the expression flipping the fraction and changing the sign of the exponent: (4/9)^-3 = (9/4)^3.
Now, raise each factor in the fraction to the power of 3: 9^3/4^3.
Finally, evaluate the expression: 9^3 = 729, and 4^3 = 64. So the final answer is 729/64.
4. [(1-4)^-3 Divide (1/4)^2]^-2:
To simplify this expression, let's break it down step by step:
First, simplify the expression inside the brackets: (1-4)^-3 Divide (1/4)^2 = (-3)^-3 Divide (1/16).
Now, let's work on (-3)^-3. To raise a negative number to a negative exponent, you can first convert the negative exponent to a positive exponent by taking the reciprocal of the base raised to the positive version of the exponent. So (-3)^-3 becomes 1/(-3)^3 or -1/(3)^3.
Next, evaluate the expression: -1/(3)^3 = -1/27.
Now, substitute this result back into the main expression: -1/27 Divide (1/16).
To divide fractions, you can multiply by the reciprocal of the second fraction. So the expression becomes -1/27 * 16/1.
Multiply the numerators and denominators together: -16/27.
Finally, raise the expression to the power of -2. To do that, you can take the reciprocal and square it. So -16/27 becomes (-27/16)^2.
The final answer is (-27/16)^2.
1. (3^3)(3^-6)
To simplify this expression with positive exponents, we can subtract the exponents:
3^3 * 3^-6 = 3^(3 - 6) = 3^-3
2. S^2 t^-1 / (s^4 t^4)^1/2
Let's simplify the expression step-by-step:
First, let's simplify the denominator:
(s^4 t^4)^(1/2) = s^(4 * 1/2) * t^(4 * 1/2) = s^2 * t^2
Now, we can rewrite the expression as:
S^2 t^-1 / (s^2 t^2)
Next, let's simplify the exponents:
t^-1 = 1/t
Substituting this back into the expression, we get:
S^2 / (s^2 t)
3. [(2/3)^2]^-3
To simplify this expression, let's first evaluate the inner exponent:
(2/3)^2 = (2^2) / (3^2) = 4/9
Now, let's apply the outer exponent:
(4/9)^-3 = (9/4)^3 = 729/64 ≈ 11.3906
4. [(1-4)^-3 / (1/4)^2]^-2
To simplify this expression, let's evaluate the inner exponents:
(1-4)^-3 = (-3)^-3 = (-1/3)^3 = -1/27
and
(1/4)^2 = (1^2) / (4^2) = 1/16
Now, let's divide the two results:
(-1/27) / (1/16) = (-1/27) * (16/1) = -16/27 ≈ -0.5926
Finally, let's apply the outer exponent:
(-16/27)^-2 = (27/16)^2 = 729/256 ≈ 2.8477