Solved on Feb 04, 2024

Simplify the expression $3^{-3} \div 3^{-4} \cdot 9^{-2}$ into an integer or simplified fraction.

STEP 1

Assumptions
1. We are working with exponential expressions involving the base number 3.
2. Division and multiplication of exponents follow the laws of exponents.
3. We aim to simplify the expression to an integer or a fraction in its simplest form.

STEP 2

Recall the law of exponents for division: $a^{m} \div a^{n} = a^{m-n}$ .

STEP 3

Apply the law of exponents to the first part of the expression $3^{-3} \div 3^{-4}$ .
$3^{-3} \div 3^{-4} = 3^{-3 - (-4)} = 3^{1}$

STEP 4

Recall that $9$ is a power of $3$ , specifically $9 = 3^2$ .

STEP 5

Rewrite $9^{-2}$ using the base number $3$ .
$9^{-2} = (3^2)^{-2}$

STEP 6

Recall the law of exponents for powers raised to powers: $(a^{m})^{n} = a^{mn}$ .

STEP 7

Apply the law of exponents to $(3^2)^{-2}$ .
$9^{-2} = (3^2)^{-2} = 3^{2 \cdot (-2)} = 3^{-4}$

STEP 8

Now we have the simplified expression with a common base of $3$ .
$3^{1} \cdot 3^{-4}$

STEP 9

Recall the law of exponents for multiplication: $a^{m} \cdot a^{n} = a^{m+n}$ .

STEP 10

Apply the law of exponents to multiply $3^{1}$ and $3^{-4}$ .
$3^{1} \cdot 3^{-4} = 3^{1 + (-4)} = 3^{-3}$

STEP 11

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent: $a^{-n} = \frac{1}{a^{n}}$ .

STEP 12

Apply the rule for negative exponents to $3^{-3}$ .
$3^{-3} = \frac{1}{3^3}$

STEP 13

Calculate $3^3$ to find the denominator of the fraction.
$3^3 = 3 \cdot 3 \cdot 3 = 27$

STEP 14

Write the final simplified expression.
$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$
The expression $3^{-3} \div 3^{-4} \cdot 9^{-2}$ simplifies to $\frac{1}{27}$ .

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Simplify the expression $3^{-3} \div 3^{-4} \cdot 9^{-2}$ into an integer or simplified fraction.

STEP 1

Assumptions
1. We are working with exponential expressions involving the base number 3.
2. Division and multiplication of exponents follow the laws of exponents.
3. We aim to simplify the expression to an integer or a fraction in its simplest form.

STEP 2

Recall the law of exponents for division: $a^{m} \div a^{n} = a^{m-n}$ .

STEP 3

Apply the law of exponents to the first part of the expression $3^{-3} \div 3^{-4}$ .
$3^{-3} \div 3^{-4} = 3^{-3 - (-4)} = 3^{1}$

STEP 4

Recall that $9$ is a power of $3$ , specifically $9 = 3^2$ .

STEP 5

Rewrite $9^{-2}$ using the base number $3$ .
$9^{-2} = (3^2)^{-2}$

STEP 6

Recall the law of exponents for powers raised to powers: $(a^{m})^{n} = a^{mn}$ .

STEP 7

Apply the law of exponents to $(3^2)^{-2}$ .
$9^{-2} = (3^2)^{-2} = 3^{2 \cdot (-2)} = 3^{-4}$

STEP 8

Now we have the simplified expression with a common base of $3$ .
$3^{1} \cdot 3^{-4}$

STEP 9

Recall the law of exponents for multiplication: $a^{m} \cdot a^{n} = a^{m+n}$ .

STEP 10

Apply the law of exponents to multiply $3^{1}$ and $3^{-4}$ .
$3^{1} \cdot 3^{-4} = 3^{1 + (-4)} = 3^{-3}$

STEP 11

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent: $a^{-n} = \frac{1}{a^{n}}$ .

STEP 12

Apply the rule for negative exponents to $3^{-3}$ .
$3^{-3} = \frac{1}{3^3}$

STEP 13

Calculate $3^3$ to find the denominator of the fraction.
$3^3 = 3 \cdot 3 \cdot 3 = 27$

STEP 14

Write the final simplified expression.
$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$
The expression $3^{-3} \div 3^{-4} \cdot 9^{-2}$ simplifies to $\frac{1}{27}$ .

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Simplify the expression 3−3÷3−4⋅9−23^{-3} \div 3^{-4} \cdot 9^{-2}3−3÷3−4⋅9−2 into an integer or simplified fraction.

STEP 1

Assumptions1. We are working with exponential expressions involving the base number 3.2. Division and multiplication of exponents follow the laws of exponents.3. We aim to simplify the expression to an integer or a fraction in its simplest form.

STEP 2

Recall the law of exponents for division: am÷an=am−na^{m} \div a^{n} = a^{m-n}am÷an=am−n.

STEP 3

Apply the law of exponents to the first part of the expression 3−3÷3−43^{-3} \div 3^{-4}3−3÷3−4.3−3÷3−4=3−3−(−4)=313^{-3} \div 3^{-4} = 3^{-3 - (-4)} = 3^{1}3−3÷3−4=3−3−(−4)=31

STEP 4

Recall that 999 is a power of 333, specifically 9=329 = 3^29=32.

STEP 5

Rewrite 9−29^{-2}9−2 using the base number 333.9−2=(32)−29^{-2} = (3^2)^{-2}9−2=(32)−2

STEP 6

Recall the law of exponents for powers raised to powers: (am)n=amn(a^{m})^{n} = a^{mn}(am)n=amn.

STEP 7

Apply the law of exponents to (32)−2(3^2)^{-2}(32)−2.9−2=(32)−2=32⋅(−2)=3−49^{-2} = (3^2)^{-2} = 3^{2 \cdot (-2)} = 3^{-4}9−2=(32)−2=32⋅(−2)=3−4

STEP 8

Now we have the simplified expression with a common base of 333.31⋅3−43^{1} \cdot 3^{-4}31⋅3−4

STEP 9

Recall the law of exponents for multiplication: am⋅an=am+na^{m} \cdot a^{n} = a^{m+n}am⋅an=am+n.

STEP 10

Apply the law of exponents to multiply 313^{1}31 and 3−43^{-4}3−4.31⋅3−4=31+(−4)=3−33^{1} \cdot 3^{-4} = 3^{1 + (-4)} = 3^{-3}31⋅3−4=31+(−4)=3−3

STEP 11

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent: a−n=1ana^{-n} = \frac{1}{a^{n}}a−n=an1​.

STEP 12

Apply the rule for negative exponents to 3−33^{-3}3−3.3−3=1333^{-3} = \frac{1}{3^3}3−3=331​

STEP 13

Calculate 333^333 to find the denominator of the fraction.33=3⋅3⋅3=273^3 = 3 \cdot 3 \cdot 3 = 2733=3⋅3⋅3=27

STEP 14

Write the final simplified expression.3−3=133=1273^{-3} = \frac{1}{3^3} = \frac{1}{27}3−3=331​=271​The expression 3−3÷3−4⋅9−23^{-3} \div 3^{-4} \cdot 9^{-2}3−3÷3−4⋅9−2 simplifies to 127\frac{1}{27}271​.

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Simplify the expression $3^{-3} \div 3^{-4} \cdot 9^{-2}$ into an integer or simplified fraction.

Assumptions
1. We are working with exponential expressions involving the base number 3.
2. Division and multiplication of exponents follow the laws of exponents.
3. We aim to simplify the expression to an integer or a fraction in its simplest form.

Recall the law of exponents for division: $a^{m} \div a^{n} = a^{m-n}$ .

Apply the law of exponents to the first part of the expression $3^{-3} \div 3^{-4}$ .
$3^{-3} \div 3^{-4} = 3^{-3 - (-4)} = 3^{1}$

Recall that $9$ is a power of $3$ , specifically $9 = 3^2$ .

Rewrite $9^{-2}$ using the base number $3$ .
$9^{-2} = (3^2)^{-2}$

Recall the law of exponents for powers raised to powers: $(a^{m})^{n} = a^{mn}$ .

Apply the law of exponents to $(3^2)^{-2}$ .
$9^{-2} = (3^2)^{-2} = 3^{2 \cdot (-2)} = 3^{-4}$

Now we have the simplified expression with a common base of $3$ .
$3^{1} \cdot 3^{-4}$

Recall the law of exponents for multiplication: $a^{m} \cdot a^{n} = a^{m+n}$ .

Apply the law of exponents to multiply $3^{1}$ and $3^{-4}$ .
$3^{1} \cdot 3^{-4} = 3^{1 + (-4)} = 3^{-3}$

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent: $a^{-n} = \frac{1}{a^{n}}$ .

Apply the rule for negative exponents to $3^{-3}$ .
$3^{-3} = \frac{1}{3^3}$

Calculate $3^3$ to find the denominator of the fraction.
$3^3 = 3 \cdot 3 \cdot 3 = 27$

Write the final simplified expression.
$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$
The expression $3^{-3} \div 3^{-4} \cdot 9^{-2}$ simplifies to $\frac{1}{27}$ .