Math / Algebra

Question6. Evaluate $5^{4} \cdot 5^{-4}$ (A) 0 (B) 1 (C) 5 (D) $5^{-16}$

Studdy Solution

STEP 1

What is this asking? We need to figure out what $5^4 \cdot 5^{-4}$ equals. Watch out! Don't fall into the trap of multiplying the exponents directly!
Remember your exponent rules.

STEP 2

1. Rewrite the negative exponent.
2. Multiply the fractions.
3. Simplify the expression.

STEP 3

A negative exponent means "one over." So, $5^{-4}$ is the same as $\frac{1}{5^4}$ .
Let's rewrite our problem with this in mind:
$5^4 \cdot 5^{-4} = 5^4 \cdot \frac{1}{5^4}$

STEP 4

Now, we multiply the fractions.
Remember, any number can be written as a fraction over one.
So, $5^4$ can be written as $\frac{5^4}{1}$ .
Now, we have:
$\frac{5^4}{1} \cdot \frac{1}{5^4}$

STEP 5

When multiplying fractions, we multiply the numerators (the top parts) together and the denominators (the bottom parts) together:
$\frac{5^4 \cdot 1}{1 \cdot 5^4} = \frac{5^4}{5^4}$

STEP 6

Now, we have a fraction where the numerator and denominator are the same.
Any number divided by itself is 1 (we're dividing to one!), unless that number is zero.
Since $5^4$ is definitely not zero, we can simplify this fraction to 1:
$\frac{5^4}{5^4} = 1$

STEP 7

Our final answer is 1, which corresponds to answer choice (B).

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Question6. Evaluate $5^{4} \cdot 5^{-4}$ (A) 0 (B) 1 (C) 5 (D) $5^{-16}$

Studdy Solution

STEP 1

What is this asking? We need to figure out what $5^4 \cdot 5^{-4}$ equals. Watch out! Don't fall into the trap of multiplying the exponents directly!
Remember your exponent rules.

STEP 2

1. Rewrite the negative exponent.
2. Multiply the fractions.
3. Simplify the expression.

STEP 3

A negative exponent means "one over." So, $5^{-4}$ is the same as $\frac{1}{5^4}$ .
Let's rewrite our problem with this in mind:
$5^4 \cdot 5^{-4} = 5^4 \cdot \frac{1}{5^4}$

STEP 4

Now, we multiply the fractions.
Remember, any number can be written as a fraction over one.
So, $5^4$ can be written as $\frac{5^4}{1}$ .
Now, we have:
$\frac{5^4}{1} \cdot \frac{1}{5^4}$

STEP 5

When multiplying fractions, we multiply the numerators (the top parts) together and the denominators (the bottom parts) together:
$\frac{5^4 \cdot 1}{1 \cdot 5^4} = \frac{5^4}{5^4}$

STEP 6

Now, we have a fraction where the numerator and denominator are the same.
Any number divided by itself is 1 (we're dividing to one!), unless that number is zero.
Since $5^4$ is definitely not zero, we can simplify this fraction to 1:
$\frac{5^4}{5^4} = 1$

STEP 7

Our final answer is 1, which corresponds to answer choice (B).

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Question6. Evaluate 54⋅5−45^{4} \cdot 5^{-4}54⋅5−4 (A) 0 (B) 1 (C) 5 (D) 5−165^{-16}5−16

Studdy Solution

STEP 1

What is this asking? We need to figure out what 54⋅5−45^4 \cdot 5^{-4}54⋅5−4 equals. Watch out! Don't fall into the trap of multiplying the exponents directly!Remember your exponent rules.

STEP 2

1. Rewrite the negative exponent.2. Multiply the fractions.3. Simplify the expression.

STEP 3

A negative exponent means "one over." So, 5−45^{-4}5−4 is the same as 154\frac{1}{5^4}541​.Let's rewrite our problem with this in mind:54⋅5−4=54⋅1545^4 \cdot 5^{-4} = 5^4 \cdot \frac{1}{5^4}54⋅5−4=54⋅541​

STEP 4

Now, we multiply the fractions.Remember, any number can be written as a fraction over one.So, 545^454 can be written as 541\frac{5^4}{1}154​.Now, we have:541⋅154\frac{5^4}{1} \cdot \frac{1}{5^4}154​⋅541​

STEP 5

When multiplying fractions, we multiply the numerators (the top parts) together and the denominators (the bottom parts) together:54⋅11⋅54=5454\frac{5^4 \cdot 1}{1 \cdot 5^4} = \frac{5^4}{5^4}1⋅5454⋅1​=5454​

STEP 6

Now, we have a fraction where the numerator and denominator are the same.Any number divided by itself is **1** (we're dividing to one!), unless that number is zero.Since 545^454 is definitely not zero, we can simplify this fraction to **1**:5454=1\frac{5^4}{5^4} = 15454​=1

STEP 7

Our final answer is **1**, which corresponds to answer choice (B).

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Question6. Evaluate $5^{4} \cdot 5^{-4}$ (A) 0 (B) 1 (C) 5 (D) $5^{-16}$

What is this asking? We need to figure out what $5^4 \cdot 5^{-4}$ equals. Watch out! Don't fall into the trap of multiplying the exponents directly!
Remember your exponent rules.

1. Rewrite the negative exponent.
2. Multiply the fractions.
3. Simplify the expression.

A negative exponent means "one over." So, $5^{-4}$ is the same as $\frac{1}{5^4}$ .
Let's rewrite our problem with this in mind:
$5^4 \cdot 5^{-4} = 5^4 \cdot \frac{1}{5^4}$

Now, we multiply the fractions.
Remember, any number can be written as a fraction over one.
So, $5^4$ can be written as $\frac{5^4}{1}$ .
Now, we have:
$\frac{5^4}{1} \cdot \frac{1}{5^4}$

When multiplying fractions, we multiply the numerators (the top parts) together and the denominators (the bottom parts) together:
$\frac{5^4 \cdot 1}{1 \cdot 5^4} = \frac{5^4}{5^4}$

Now, we have a fraction where the numerator and denominator are the same.
Any number divided by itself is 1 (we're dividing to one!), unless that number is zero.
Since $5^4$ is definitely not zero, we can simplify this fraction to 1:
$\frac{5^4}{5^4} = 1$

Our final answer is 1, which corresponds to answer choice (B).