Drag each tile to the correct box. Not all tiles will be used. Drag the explanations used to solve the given equation in the correct sequence. [tex]5^{(6 x-9)}=125[/tex]
* subtract 9 from each side
* divide each side by 6
* add 9 to each side
* rewrite 125 as a power of 5
* equate the exponents of the equation
* divide each side by 5
Answer (2)
Rewrite 125 as 5 3 , so the equation becomes 5 ( 6 x − 9 ) = 5 3 .
Equate the exponents: 6 x − 9 = 3 .
Add 9 to both sides: 6 x = 12 .
Divide by 6 to find the solution: x = 2 .
Explanation
Understanding the Problem We are given the equation 5 ( 6 x − 9 ) = 125 and we need to solve for x . The goal is to isolate x by performing algebraic operations on both sides of the equation.
Rewrite 125 as a power of 5 First, we rewrite 125 as a power of 5. Since 125 = 5 × 5 × 5 = 5 3 , we can rewrite the equation as 5 ( 6 x − 9 ) = 5 3 .
Equate the exponents of the equation Now, we equate the exponents of the equation. Since the bases are equal (both are 5), we can set the exponents equal to each other: 6 x − 9 = 3 .
Add 9 to each side Next, we add 9 to each side of the equation to isolate the term with x : 6 x − 9 + 9 = 3 + 9 , which simplifies to 6 x = 12 .
Divide each side by 6 Finally, we divide each side by 6 to solve for x : 6 6 x = 6 12 , which simplifies to x = 2 .
Examples
Exponential equations are used in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account with compound interest, the amount of money you have after a certain time can be modeled using an exponential equation. Solving these equations helps you determine how long it will take for your investment to reach a certain value, or how much interest you need to earn to achieve your financial goals. Understanding exponential equations is crucial for making informed decisions in finance, science, and many other fields.
To solve 5 ( 6 x − 9 ) = 125 , we first rewrite 125 as 5 3 , equate the exponents, and then isolate x by performing algebraic operations. The steps include adding 9 to both sides and dividing by 6 to find x = 2 . The correct order of explanations includes rewriting the power, equating exponents, adding 9, and dividing by 6.
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