Apply All Exponent Rules Practice

To simplify the expression 37⋅3−9 with positive exponents, we can use the following exponent rules:

1. Product of powers rule: am * an = am+n
2. Power of a power rule: (am)n = am*n
3. Negative exponent rule: a^-n = 1/an

Let's simplify the expression step by step:

37⋅3−9

First, let's apply the negative exponent rule to get rid of the negative exponent:

37⋅3−9 = 37 * 1/3^9

Next, let's simplify the fraction by applying the power of a power rule:

37 * 1/3^9 = 37 * 1/(3^3)^3

Now, let's apply the power of a power rule:

37 * 1/(3^3)^3 = 37 * 1/3^27

Finally, let's simplify the expression:

37 * 1/3^27 = 37/3^27

So, the simplified expression is 37/3^27.

37⋅3−9=___

To simplify 37⋅3−9, we can use the rule:

a^(-n) = 1/(a^n)

Applying this rule to the expression, we have:

37⋅3^(-9) = 37/(3^9)

Thus, the value of 37⋅3−9 is 37/(3^9).

1. 3^7*3^-9= 1/9

2.2^2/2^-4= 64
3.4^0*5^-2/(-2)^-3= 8/25
4.(14^3*27^-7)^0/3^4= 1/81
5.(4^3*7^-2)^-5/7^9*4^-13= 7/16

To apply the properties of integer exponents and generate equivalent expressions to 37⋅3−9 with only positive exponents, we can use the following exponent rules:

1. Product of Powers Rule: am⋅an = am+n (where a is a non-zero number and m and n are integers)
2. Quotient of Powers Rule: am/an = am-n (where a is a non-zero number and m and n are integers)
3. Power of a Power Rule: (am)n = am⋅n (where a is a non-zero number and m and n are integers)

Let's apply these rules step-by-step:

Step 1: Rewrite 37 as a base and exponent of 3:
37 = 3^7

Step 2: Apply the Product of Powers Rule:
37⋅3^-9 = (3^7)⋅3^-9

Step 3: Combine the exponents (7+(-9)) using the Quotient of Powers Rule:
(3^7)⋅3^-9 = 3^(7+(-9))

Step 4: Simplify the exponent (7+(-9)) to -2:
3^(7+(-9)) = 3^-2

Step 5: Rewrite 3^-2 as a fraction with a positive exponent:
3^-2 = 1/3^2

Step 6: Simplify 1/3^2:
1/3^2 = 1/9

Therefore, the expression 37⋅3^-9 is equivalent to 1/9.

The solution is 1/9.

To apply the properties of integer exponents, we need to understand the different rules involved. Here are the rules we will use:

1. Product Rule: For any numbers a and b, and any integer n, a^n * b^n = (a * b)^n.
2. Power Rule: For any number a and any integers m and n, (a^m)^n = a^(m * n).
3. Quotient Rule: For any numbers a and b, and any integer n, a^n / b^n = (a/b)^n.
4. Negative Exponent Rule: For any nonzero number a, a^(-n) = 1 / a^n.
5. Zero Exponent Rule: For any nonzero number a, a^0 = 1.

Now let's apply these rules to generate equivalent expressions for 37⋅3^(-9) with only positive exponents:

Step 1: Apply the product rule.
37 * 3^(-9) = (37 * 3)^(-9)

Step 2: Simplify the expression inside the parentheses.
37 * 3 = 111
Therefore, (37 * 3)^(-9) = 111^(-9)

Step 3: Apply the negative exponent rule.
111^(-9) = 1 / 111^9

Now we have an equivalent expression with only positive exponents. To solve this expression further, we can evaluate 111^9 and write the answer as a simplified fraction with no remaining exponents.