Rewrite the following expressions: 27^1/2⋅27^1/4.
27^1/8
27^3/4
27−−√4
27−−√3
(27^(1/2)) * (27^(1/4)) = 27^(1/2 + 1/4) = 27^(3/4)
√(27^4) = √(27^2 * 27^2) = 27^(2/2) = 27^1 = 27
√(27^3) = 27^(3/2)
Which option is right bot?
Apologies for the confusion. The correct answer is (27^(1/2)) * (27^(1/4)) = 27^(1/2 + 1/4) = 27^(3/4).
So, (27^(1/2)) * (27^(1/4)) is equal to 27^(3/4).
What is an equivalent expression for −(b^1/4)^12?
1/b^3
−b^1/3
−b^−3
−b^3
To find an equivalent expression for −(b^(1/4))^12, we can simplify the given expression step by step.
First, we apply the exponent rule of raising a power to another power. (b^(1/4))^12 is equal to b^(12 * 1/4) = b^(3/1) = b^3.
Next, we apply the negative sign in front of the expression, which gives us −b^3.
Therefore, an equivalent expression for −(b^(1/4))^12 is −b^3.
Select the correct answer to the following equation: 27x^3/4−1=26.
25/27
1
3
0
To solve the equation 27x^(3/4) - 1 = 26, we need to isolate the x-term. We can do this by adding 1 to both sides of the equation:
27x^(3/4) - 1 + 1 = 26 + 1
27x^(3/4) = 27
Next, we divide both sides of the equation by 27 to solve for x:
(27x^(3/4))/27 = 27/27
x^(3/4) = 1
To remove the fractional exponent, we raise both sides of the equation to the 4/3 power:
(x^(3/4))^(4/3) = 1^(4/3)
x = 1
Thus, the correct answer to the equation 27x^(3/4) - 1 = 26 is x = 1.
Is the equation (8–√4)^−24⋅(8–√)^16=8^−48 true or false? Select the response that correctly answers the question and provides the appropriate justification.
True, but the equation can be simplified further.
True, but the equation can be simplified further.
False, because the exponents should be added rather than multiplied.
False, because the exponents should be added rather than multiplied.
True. When the bases are multiplied, the exponents are also multiplied.
True. When the bases are multiplied, the exponents are also multiplied.
False, because the Negative Exponent Rule should be applied.
False, because the Negative Exponent Rule should be applied.
The correct answer is:
False, because the exponents should be added rather than multiplied.
To simplify the equation, we can first simplify the individual terms:
(8 - √4)^-24 = (8 - 2)^-24 = 6^-24
(8 - √4)^16 = (8 - 2)^16 = 6^16
Then, we can rewrite the equation as:
6^-24 * 6^16 = 8^-48
To combine the terms on the left side, we add the exponents:
6^(-24 + 16) = 8^-48
6^-8 = 8^-48
Since the bases on both sides are different, the equation is not true. Therefore, the correct response is:
False, because the exponents should be added rather than multiplied.
Is the equation 7^−4⋅7^3/49√=7 true or false? Select the response that correctly answers the question and provides the appropriate justification.
True, because the Quotient Property gives a final exponent of 1.
True, because the Quotient Property gives a final exponent of 1.
True, because the Product Property makes the numerator and the denominator equal.
True, because the Product Property makes the numerator and the denominator equal.
False. When applying the Quotient Property, the final exponent is not 1.
False. When applying the Quotient Property, the final exponent is not 1.
False. The answer should be negative since the exponents are negative.
The correct answer is:
True, because the Quotient Property gives a final exponent of 1.
To simplify the equation, we apply the Quotient Property of exponents which states that when dividing with the same base, we subtract the exponents:
7^(-4) * 7^(3/49√) = 7^(-4 + 3/49√)
To simplify further, we need to find a common denominator for the exponents:
-4 + 3/49√ = -4 + (3/49)(1/√)
Next, we can combine the terms by finding a common denominator:
-4 + (3/49)(1/√) = (-4 * 49 + 3/√) / 49
Simplifying the numerator gives us:
(-196 + 3/√) / 49
Since 3/√ is equal to 3√, the expression becomes:
(-196 + 3√) / 49
Now, since 3√ is equal to 3 times the square root of 7, the expression becomes:
(-196 + 3√) / 49 = (-196 + 3√) / 49
Therefore, the equation 7^(-4) * 7^(3/49√) = 7 is true, as the numerator and denominator remain equal.
Consider the expression 47/3+121−−−√. Will the sum of the two rational numbers produce a rational number?
Yes, the sum of two rational numbers will always produce a rational number. In the given expression, (47/3) + (121 − √) involves the addition of two rational numbers, 47/3 and 121. Both 47/3 and 121 are rational numbers because they can be expressed as fractions or whole numbers. Therefore, their sum will also be a rational number.
Prove that the product of the two rational numbers 36−−√ and 5/3 is a rational number. Find the product and explain how you know it is a rational number.
To prove that the product of the two rational numbers 36−√ and 5/3 is a rational number, we need to show that the product can be written as a fraction in the form a/b, where a and b are integers and b is not equal to 0.
First, let's find the product of the two numbers:
Product = (36−√) * (5/3)
We can simplify this expression by multiplying the numerators together and multiplying the denominators together:
Product = (36 * 5) / (√ * 3)
The numerator simplifies to:
36 * 5 = 180
Now, let's consider the denominator. Since √ denotes the square root, we can express it as a rational number. The square root of any positive integer can be written as a fraction in the form of √n = √n/1.
Therefore, the expression (√ * 3) can be written as:
3 * √ = 3 * √1 = 3 * (√1/1)
Now, let's rewrite our product:
Product = 180 / (3 * (√1/1))
Simplifying further, we have:
Product = 180 / (3 * 1)
Product = 180 / 3
Product = 60
We have found that the product of 36−√ and 5/3 is equal to 60, which is an integer. Since an integer can be expressed as a fraction with a denominator of 1, this confirms that the product is a rational number.