Question

help i need to complete this math problem but I don't quite understand it can someone give me the answer

Answer 1

Answer:

$x=4,\ y=3\text{ and }z=7$

Step-by-step explanation:

$\text{We need to use the laws of indices to solve this question.}$

$\text{1. }a^m\div a^n=a^{m-n}\\\text{2. }a^m\times a^n=a^{m+n}$

$\dfrac{5^{12}\times 4^4\times 3^8\times4}{4^2\times5^5\times3\times5^3}=5^x\times4^y\times3^z\\\\\\\text{or, }\dfrac{5^{12}\times4^{4+1}\times3^8}{4^2\times5^{5+3}\times3}=5^x\times4^y\times3^z\\\\\\\text{or, }\dfrac{5^{12}\times4^5\times3^8}{5^8\times4^2\times3}=5^x\times4^y\times3^z$

$\text{or, }5^{12-8}\times4^{5-2}\times3^{8-1}=5^x\times4^y\times3^z\\\\\text{or, }5^4\times4^3\times3^7=5^x\times4^y\times3^z\\\\\text{Equating powers of common bases, }\\\\x=4,\ y=3\ \text{and }z=7$

Answer 2

Answer:

x = 4

y = 3

z = 7

Step-by-step explanation:

Given equation:

$\dfrac{5^{12}\times 4^4 \times 3^8 \times 4}{4^2 \times 5^5 \times 3 \times 5^3}=5^x \times 4^y \times 3^z$

"Pronumerals" are symbols or letters used in mathematical expressions or equations as placeholders for unknown values.

To find the values of the pronumerals x, y and z in the given equation, we can use the Laws of Exponents to simplify the left side of the equation.

$\boxed{\begin{array}{c}\underline{\textsf{Laws of Exponents}}\\\\\textsf{Product:}\;\;a^m \times a^n=a^{m+n}\\\\\textsf{Quotient:}\;\;\dfrac{a^m}{a^n}=a^{m-n}\end{array}}$

Begin by applying the product law of exponents by adding the exponents of the terms with the same base in the numerator and denominator:

$\dfrac{5^{12}\times 4^{4+1} \times 3^8}{4^2 \times 5^{5+3} \times 3}=5^x \times 4^y \times 3^z\\\\\\\\\dfrac{5^{12}\times 4^{5} \times 3^8}{4^2 \times 5^{8} \times 3}=5^x \times 4^y \times 3^z$

According to the commutative property of multiplication, the order of the numbers being multiplied does not affect the result. Therefore, we can rearrange the order of the terms in the denominator as follows:

$\dfrac{5^{12}\times 4^{5} \times 3^8}{5^{8} \times 4^2 \times 3}=5^x \times 4^y \times 3^z$

The quotient property of division states that the division of two products is equal to the product of their respective quotients. Therefore, we can rewrite the left side of the equation as:

$\dfrac{5^{12}}{5^8} \times \dfrac{4^5}{4^2}\times \dfrac{3^8}{3}=5^x \times 4^y \times 3^z$

Now, apply the quotient law of exponents by subtracting the exponents of the terms of the quotients with the same base:

$5^{12-8}\times 4^{5-2}\times 3^{8-1}=5^x \times 4^y \times 3^z\\\\\\5^{4}\times 4^{3}\times 3^{7}=5^x \times 4^y \times 3^z$

Comparing the exponents of the left and right sides of the equation, the values of x, y and z are:

$\Large\text{ x=\boxed{4} }$

$\Large\text{ y=\boxed{3} }$

$\Large\text{ z=\boxed{7} }$