QuestionSolve the equations: $8^{x-1}=\sqrt[3]{2^{x-3}}$ and $3 \cdot 5^{x}+5^{x+1}=8 \cdot 5^{3}$ .

Studdy Solution

STEP 1

Assumptions1. We are given two equations, $8^{x-1}=\sqrt[3]{^{x-3}}$ and $3 \cdot5^{x}+5^{x+1}=8 \cdot5^{3}$ . . We are assuming that the base of the exponentials and radicals are positive real numbers.
3. We are assuming that the exponent x is a real number.

STEP 2

Let's start with the first equation $8^{x-1}=\sqrt[]{2^{x-}}$ . We can rewrite8 and2 in terms of the same base to simplify the equation.
8 =2^So, the equation becomes(2^)^{x-1}=\sqrt[]{2^{x-}}

STEP 3

Apply the power of a power rule in the left side of the equation. This rule states that $(a^m)^n = a^{mn}$ .
$2^{3(x-1)}=\sqrt[3]{2^{x-3}}$

STEP 4

The cube root can be written as a power of one-third. So, the equation becomes $2^{3(x-1)}=2^{\frac{x-3}{3}}$

STEP 5

Since the bases are the same, we can set the exponents equal to each other. This gives us our first equation in x.
$3(x-1)=\frac{x-3}{3}$

STEP 6

Now, let's solve the equation for x. First, multiply both sides of the equation by3 to get rid of the fraction.
$9(x-1)=x-3$

STEP 7

Expand the left side and simplify the equation.
$9x -9 = x -3$

STEP 8

Subtract x from both sides of the equation.
$8x - = -3$

STEP 9

Add9 to both sides of the equation.
$8x =6$

STEP 10

Finally, divide both sides of the equation by8 to solve for x.
$x = \frac{6}{8} = \frac{3}{4}$

STEP 11

Now, let's solve the second equation $3 \cdot5^{x}+5^{x+}=8 \cdot5^{3}$ . We can simplify the equation by dividing both sides by $5^3$ .
$\frac{3 \cdot5^{x}}{5^3}+\frac{5^{x+}}{5^3}=8$

STEP 12

implify the equation using the rule $a^{m-n} = \frac{a^m}{a^n}$ .
$\cdot5^{x-}+5^{x-}=8$

STEP 13

Since $x = \frac{3}{}$ , substitute $\frac{3}{}$ for x in the equation.
$3 \cdot5^{\frac{3}{}-3}+5^{\frac{3}{}-3}=8$

STEP 14

implify the equation.
$3 \cdot^{-\frac{9}{4}}+^{-\frac{9}{4}}=8$

STEP 15

Multiply both sides of the equation by $5^{\frac{9}{4}}$ to get rid of the negative exponent.
$3 +5^{\frac{9}{4}} =8 \cdot5^{\frac{9}{4}}$

STEP 16

Subtract $5^{\frac{9}{4}}$ from both sides of the equation.
$3 =8 \cdot5^{\frac{9}{4}} -5^{\frac{9}{4}}$

STEP 17

Factor out $5^{\frac{9}{4}}$ on the right side of the equation.
$3 =5^{\frac{9}{4}}( -)$

STEP 18

implify the right side of the equation.
$3 =7 \cdot5^{\frac{}{4}}$

STEP 19

Divide both sides of the equation by7.
$\frac{3}{7} =5^{\frac{9}{4}}$

STEP 20

Take the fourth root of both sides of the equation to solve for $5^{\frac{9}{4}}$ .
$\sqrt[4]{\frac{3}{7}} =5^{\frac{9}{16}}$

STEP 21

Raise both sides of the equation to the power of16 to solve for5.
$\left(\sqrt[4]{\frac{3}{7}}\right)^{16} =5^9$

STEP 22

Calculate the left side of the equation.
$\left(\sqrt[4]{\frac{}{7}}\right)^{16} = \frac{81}{823543}$

STEP 23

Set the equation equal to5.
$\frac{81}{823543} =5$

STEP 24

This equation is not true, so there is no solution to the system of equations.

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