Solved on Sep 18, 2023

Find the value of $d$ in the equation $5^{6} \times \sqrt{5}=5^{d}$ . Give the answer as a whole number or simplified fraction.

STEP 1

Assumptions1. We are given the equation $5^{6} \times \sqrt{5}=5^{d}$ . We need to find the value of $d$
3. The base of the exponential and the square root is54. We know that $\sqrt{5}$ can be written as $5^{1/}$

STEP 2

First, we rewrite the square root of5 in exponential form.
$\sqrt{5} =5^{1/2}$

STEP 3

Now, we substitute $5^{1/2}$ for $\sqrt{5}$ in the original equation.
$5^{6} \times5^{1/2} =5^{d}$

STEP 4

Next, we use the rule of exponents that says when you multiply two exponents with the same base, you add the exponents.
$^{6+1/2} =^{d}$

STEP 5

Calculate the sum of the exponents.
$5^{.5} =5^{d}$

STEP 6

Since the bases are the same, the exponents must be equal for the equation to hold true. Therefore, we can equate the exponents.
$6.5 = d$

STEP 7

We can write the decimal6.5 as a fraction in its simplest form.
$d =13/2$ The value of $d$ is $13/2$ .

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Find the value of $d$ in the equation $5^{6} \times \sqrt{5}=5^{d}$ . Give the answer as a whole number or simplified fraction.

STEP 1

Assumptions1. We are given the equation $5^{6} \times \sqrt{5}=5^{d}$ . We need to find the value of $d$
3. The base of the exponential and the square root is54. We know that $\sqrt{5}$ can be written as $5^{1/}$

STEP 2

First, we rewrite the square root of5 in exponential form.
$\sqrt{5} =5^{1/2}$

STEP 3

Now, we substitute $5^{1/2}$ for $\sqrt{5}$ in the original equation.
$5^{6} \times5^{1/2} =5^{d}$

STEP 4

Next, we use the rule of exponents that says when you multiply two exponents with the same base, you add the exponents.
$^{6+1/2} =^{d}$

STEP 5

Calculate the sum of the exponents.
$5^{.5} =5^{d}$

STEP 6

Since the bases are the same, the exponents must be equal for the equation to hold true. Therefore, we can equate the exponents.
$6.5 = d$

STEP 7

We can write the decimal6.5 as a fraction in its simplest form.
$d =13/2$ The value of $d$ is $13/2$ .

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Find the value of ddd in the equation 56×5=5d5^{6} \times \sqrt{5}=5^{d}56×5​=5d. Give the answer as a whole number or simplified fraction.

STEP 1

Assumptions1. We are given the equation 56×5=5d5^{6} \times \sqrt{5}=5^{d}56×5​=5d . We need to find the value of ddd3. The base of the exponential and the square root is54. We know that 5\sqrt{5}5​ can be written as 51/5^{1/}51/

STEP 2

First, we rewrite the square root of5 in exponential form.5=51/2\sqrt{5} =5^{1/2}5​=51/2

STEP 3

Now, we substitute 51/25^{1/2}51/2 for 5\sqrt{5}5​ in the original equation.56×51/2=5d5^{6} \times5^{1/2} =5^{d}56×51/2=5d

STEP 4

Next, we use the rule of exponents that says when you multiply two exponents with the same base, you add the exponents.6+1/2=d^{6+1/2} =^{d}6+1/2=d

STEP 5

Calculate the sum of the exponents.5.5=5d5^{.5} =5^{d}5.5=5d

STEP 6

Since the bases are the same, the exponents must be equal for the equation to hold true. Therefore, we can equate the exponents.6.5=d6.5 = d6.5=d

STEP 7

We can write the decimal6.5 as a fraction in its simplest form.d=13/2d =13/2d=13/2The value of ddd is 13/213/213/2.

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Find the value of $d$ in the equation $5^{6} \times \sqrt{5}=5^{d}$ . Give the answer as a whole number or simplified fraction.

Assumptions1. We are given the equation $5^{6} \times \sqrt{5}=5^{d}$ . We need to find the value of $d$
3. The base of the exponential and the square root is54. We know that $\sqrt{5}$ can be written as $5^{1/}$

First, we rewrite the square root of5 in exponential form.
$\sqrt{5} =5^{1/2}$

Now, we substitute $5^{1/2}$ for $\sqrt{5}$ in the original equation.
$5^{6} \times5^{1/2} =5^{d}$

Next, we use the rule of exponents that says when you multiply two exponents with the same base, you add the exponents.
$^{6+1/2} =^{d}$

Calculate the sum of the exponents.
$5^{.5} =5^{d}$

Since the bases are the same, the exponents must be equal for the equation to hold true. Therefore, we can equate the exponents.
$6.5 = d$

We can write the decimal6.5 as a fraction in its simplest form.
$d =13/2$ The value of $d$ is $13/2$ .