Solve the expression 15⋅8^0/5+4(7^0−6)
We'll follow the order of operations (also known as PEMDAS) to solve this expression:
First, we start by evaluating the exponent.
8^0 = 1 (Any number raised to the power of 0 is 1)
Now let's calculate the expression step by step:
15⋅1/5 + 4(1 − 6)
Next, let's simplify inside each parenthesis:
15⋅1/5 + 4(-5)
Now, let's multiply:
3 + (-20)
Finally, let's add:
3 - 20 = -17
Therefore, the solution to the expression 15⋅8^0/5+4(7^0−6) is -17.
Use the Power Rule of Exponents to simplify (10^3)^6 .
To simplify (10^3)^6 using the Power Rule of Exponents, we multiply the exponents.
The Power Rule of Exponents states that for any positive numbers a and b and any integers m and n, (a^m)^n = a^(m * n).
Applying this rule to our expression:
(10^3)^6 = 10^(3 * 6) = 10^18.
Therefore, the expression (10^3)^6 simplifies to 10^18.
Use the Power Rule of Exponents to create a simplified equivalent expression for (194)9 .
To create a simplified equivalent expression for (194)9 using the Power Rule of Exponents, we raise the base 194 to the power of 9.
According to the Power Rule of Exponents, for any positive number a and any integers m and n, (a^m)^n = a^(m * n).
Applying this rule to our expression:
(194)^9 = 194^(1 * 9) = 194^9.
Therefore, the simplified equivalent expression for (194)9 is 194^9.
Use the Power Rule of Exponents to create a simplified equivalent expression for (19^4)^9 .
To create a simplified equivalent expression for (19^4)^9 using the Power Rule of Exponents, we multiply the exponents.
According to the Power Rule of Exponents, for any positive number a and any integers m and n, (a^m)^n = a^(m * n).
In this case, we have (19^4)^9.
We multiply the exponents:
(19^4)^9 = 19^(4 * 9) = 19^36.
Therefore, the simplified equivalent expression for (19^4)^9 is 19^36.
Apply the Property of Negative Integer Exponents to solve 7−3 . The answer should not contain any exponent expression or decimals.(1 point)
The Property of Negative Integer Exponents states that for any non-zero number a and any negative integer n, a^(-n) = 1 / a^n.
Applying this property to our expression:
7^(-3) = 1 / 7^3.
Now, let's simplify 7^3:
7^3 = 7 * 7 * 7 = 343.
Therefore, 7^(-3) = 1 / 343.
Hence, the solution to 7^(-3) in terms of fractions and without any exponent expression or decimals is 1/343.
Apply the Property of Negative Integer Exponents to solve 7^−3 . The answer should not contain any exponent expression or decimals.(1 point)
The Property of Negative Integer Exponents states that for any non-zero number a and any negative integer n, a^(-n) = 1 / a^n.
Applying this property to our expression:
7^(-3) = 1 / 7^3.
Now, let's simplify 7^3:
7^3 = 7 * 7 * 7 = 343.
Therefore, 7^(-3) = 1 / 343.
Hence, the solution to 7^(-3) in terms of fractions and without any exponent expression or decimals is 1/343.
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3^−17/8^−14 with positive exponents only.(1 point)
3−178−14=
The Property of Negative Integer Exponents states that for any non-zero number a and any negative integer n, a^(-n) = 1 / a^n.
Applying this property to our expression:
3^(-17) / 8^(-14).
To make the exponents positive, we can invert the fractions:
(1 / 3^17) / (1 / 8^14).
Now, when we divide fractions, we multiply by the reciprocal of the second fraction:
(1 / 3^17) * (8^14 / 1).
Simplifying further, we use the Product Rule of Exponents, which states that for any positive numbers a and b, and any integers m and n, (a^m) * (b^n) = (a * b)^(m + n):
(1 * 8^14) / 3^17 = (8^14 / 3^17).
Therefore, an expression equivalent to 3^(-17) / 8^(-14) with positive exponents only is 8^14 / 3^17.
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3^−17/8^−14 with positive exponents only.(1 point)
To generate an expression equivalent to 3^(-17) / 8^(-14) with positive exponents only, we can apply the Property of Negative Integer Exponents.
According to this property, for any non-zero number a and any negative integer n, a^(-n) can be rewritten as 1 / a^n.
Applying this property to our expression:
3^(-17) / 8^(-14) = 1 / (3^17) / 1 / (8^14).
Now, when dividing fractions, we can multiply by the reciprocal of the denominator:
1 / (3^17) * (8^14 / 1).
Simplifying further, we multiply the numerators and the denominators:
8^14 / 3^17.
Therefore, an expression equivalent to 3^(-17) / 8^(-14) with positive exponents only is 8^14 / 3^17.
Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)^7 x 7^20/6^−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.
To generate equivalent expressions with only positive exponents, we can use the Properties of Integer Exponents.
First, let's apply the property (a^m)^n = a^(m*n) to simplify (7^(-3))^7:
(7^(-3))^7 = 7^(-3*7) = 7^(-21).
Next, let's apply the property a^(-n) = 1/a^n to simplify 7^(-21):
7^(-21) = 1/(7^21).
Now let's simplify the second part of the expression, 7^20/6^(-2), by applying the property a^(-n) = 1/a^n:
7^20/6^(-2) = 7^20/(1/6^2) = 7^20 * 6^2.
Now we have the simplified expression:
(7^(-3))^7 * 7^20/6^(-2) = (1/(7^21)) * 7^20 * 6^2.
To solve the expression, we can combine the terms with the same base by applying the properties of exponent arithmetic:
(1/(7^21)) * 7^20 * 6^2 = 6^2/7^21 * 7^20.
Now, we can simplify this expression:
6^2/7^21 * 7^20 = 6^2 * 7^(20-21) = 6^2 * 7^(-1) = 36/7.
Therefore, the solution to the expression (7^(-3))^7 * 7^20/6^(-2) with positive exponents is 36/7, which is an improper fraction.
Apply the product rule of integer exponents to find the numerical equivalent of 12^−5 x 12^7 . Show your work and explain how you used the product rule of integer exponents.(2 points)
The product rule of integer exponents states that for any non-zero number a and integers m and n:
a^m * a^n = a^(m + n)
In this case, we have the expression 12^(-5) * 12^7.
To find the numerical equivalent, we can apply the product rule of integer exponents.
First, let's simplify the exponents:
12^(-5) * 12^7 = 1 / 12^5 * 12^7
Now, let's apply the product rule of integer exponents by adding the exponents:
1 / 12^5 * 12^7 = 1 / 12^(5 + 7) = 1 / 12^12
Therefore, the numerical equivalent of 12^(-5) * 12^7 is 1 / 12^12.