Find the value of m in $\left(\frac{2}{7}\right)^{-3} \times \left(\frac{2}{7}\right)^{-5}=\left (\frac{2}{7}\right)^{-3m+1}$

Question

Prepp Team · Accepted Answer

Solving Exponent Equations: Finding the Value of m

This question asks us to find the value of the variable 'm' in an equation involving exponents with the same base. The equation is given as:

$ \left(\frac{2}{7}\right)^{-3} \times \left(\frac{2}{7}\right)^{-5}=\left (\frac{2}{7}\right)^{-3m+1} $

To solve this equation, we need to use the properties of exponents. A key property is the rule for multiplying exponential terms with the same base:

Product Rule of Exponents: When multiplying exponents with the same base, you add the powers. Mathematically, this is expressed as $ a^x \times a^y = a^{x+y} $.

Applying the Product Rule

Let's apply the product rule to the left side of the given equation. The base is $ \frac{2}{7} $ and the exponents are $-3$ and $-5$.

$ \left(\frac{2}{7}\right)^{-3} \times \left(\frac{2}{7}\right)^{-5} = \left(\frac{2}{7}\right)^{-3 + (-5)} $

Simplify the exponent on the left side:

$ -3 + (-5) = -3 - 5 = -8 $

So, the left side of the equation simplifies to:

$ \left(\frac{2}{7}\right)^{-8} $

Now, the original equation becomes:

$ \left(\frac{2}{7}\right)^{-8} = \left (\frac{2}{7}\right)^{-3m+1} $

Equating the Exponents

We now have an equation where both sides have the same base, $ \frac{2}{7} $. When $ a^x = a^y $ and the base $ a $ is not equal to 0, 1, or -1, we can conclude that the exponents must be equal, i.e., $ x = y $.

In our equation, the base is $ \frac{2}{7} $, which is not 0, 1, or -1. Therefore, we can equate the exponents:

$ -8 = -3m + 1 $

Solving for m

Now we have a simple linear equation to solve for 'm'. We need to isolate 'm' on one side of the equation.

Subtract 1 from both sides of the equation:

$ -8 - 1 = -3m + 1 - 1 $

$ -9 = -3m $

Now, divide both sides by -3 to find the value of 'm':

$ \frac{-9}{-3} = \frac{-3m}{-3} $

$ 3 = m $

So, the value of m is 3.

Step	Calculation / Rule	Equation
1	Original Equation	$ \left(\frac{2}{7}\right)^{-3} \times \left(\frac{2}{7}\right)^{-5}=\left (\frac{2}{7}\right)^{-3m+1} $
2	Apply $a^x \times a^y = a^{x+y}$	$ \left(\frac{2}{7}\right)^{-3 + (-5)} = \left (\frac{2}{7}\right)^{-3m+1} $
3	Simplify exponent on LHS	$ \left(\frac{2}{7}\right)^{-8} = \left (\frac{2}{7}\right)^{-3m+1} $
4	Equate exponents (since bases are equal)	$ -8 = -3m + 1 $
5	Subtract 1 from both sides	$ -9 = -3m $
6	Divide by -3	$ m = 3 $

Revision Table: Exponent Rules for Solving Equations

Rule Name	Formula	Description	Example
Product Rule	$a^x \times a^y = a^{x+y}$	To multiply powers with the same base, add their exponents.	$2^3 \times 2^4 = 2^{3+4} = 2^7$
Power of a Power Rule	$(a^x)^y = a^{xy}$	To raise a power to a power, multiply the exponents.	$(3^2)^3 = 3^{2 \times 3} = 3^6$
Quotient Rule	$ \frac{a^x}{a^y} = a^{x-y} $	To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.	$ \frac{5^6}{5^2} = 5^{6-2} = 5^4 $
Zero Exponent Rule	$a^0 = 1 \quad (a \neq 0)$	Any non-zero number raised to the power of zero is 1.	$7^0 = 1$
Negative Exponent Rule	$a^{-x} = \frac{1}{a^x}$	A term with a negative exponent is equal to its reciprocal with a positive exponent.	$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
Equating Exponents	If $a^x = a^y$ and $a \neq 0, 1, -1$, then $x=y$	If two expressions with the same base are equal, their exponents must be equal.	If $2^x = 2^5$, then $x=5$

Additional Information: Solving Exponent Equations

Solving equations involving exponents often relies on making the bases on both sides of the equation the same. Once the bases are equal, we can use the property that if $a^x = a^y$ and $a \neq 0, 1, -1$, then $x=y$. This converts the exponential equation into a simpler algebraic equation (usually linear or quadratic) that can be solved using standard techniques.

Sometimes, the bases are not initially the same, but can be expressed as powers of a common base. For example, if you have $4^x = 8$, you can rewrite this as $(2^2)^x = 2^3$, which simplifies to $2^{2x} = 2^3$. Then, you can equate the exponents: $2x = 3$, leading to $x = \frac{3}{2}$.

Remember to be careful with negative bases and exponents. The rules apply generally, but care must be taken with the domain and range of exponential functions, especially when dealing with fractional exponents or when the base is negative.

In this specific problem, the bases were already the same ($ \frac{2}{7} $), which is greater than 0 and not equal to 1, simplifying the initial steps. The primary task was to use the product rule of exponents to combine the terms on the left side.

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