Math Snap

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.
$$\frac{81}{823543} =5$$

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