$\frac{7^2 \cdot 7^8}{7^4}=\frac{7^a}{7^4}=7^b \newline a=\square, b=\square$
Answer (2)
In the equation 7 4 7 2 ⋅ 7 8 = 7 4 7 a = 7 b , we simplify the left side to find a = 10 and b = 6 . The steps include adding and subtracting exponents when multiplying and dividing powers of the same base. The final results are a = 10 , b = 6 .
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Simplify the left-hand side of the equation: 7 4 7 2 × 7 8 = 7 4 7 10 = 7 6 .
From 7 4 7 a = 7 6 , deduce that a − 4 = 6 , so a = 10 .
From 7 b = 7 6 , deduce that b = 6 .
The final answers are a = 10 and b = 6 , so a = 10 , b = 6 .
Explanation
Understanding the Problem We are given the equation 7 4 7 2 × 7 8 = 7 4 7 a = 7 b and we need to find the values of a and b .
Simplifying the Numerator First, let's simplify the left-hand side of the equation. We know that when multiplying exponential terms with the same base, we add the exponents: 7 2 × 7 8 = 7 2 + 8 = 7 10 .
Simplifying the Fraction Now, we divide 7 10 by 7 4 . When dividing exponential terms with the same base, we subtract the exponents: 7 4 7 10 = 7 10 − 4 = 7 6 .
Setting up Equations So, we have 7 6 = 7 4 7 a = 7 b . From this, we can deduce two equations. First, 7 4 7 a = 7 6 . Second, 7 b = 7 6 .
Solving for a From the equation 7 4 7 a = 7 6 , we know that 7 a − 4 = 7 6 . Therefore, a − 4 = 6 , which means a = 6 + 4 = 10 .
Solving for b From the equation 7 b = 7 6 , we can directly see that b = 6 .
Final Answer Therefore, the values of a and b are a = 10 and b = 6 .
Examples
Understanding exponential rules is crucial in many fields, such as calculating compound interest. For example, if you invest money that grows at a certain percentage annually, the exponential growth can be modeled using these rules to determine the future value of your investment. Similarly, in science, exponential decay is used to model the decay of radioactive substances, helping scientists determine the age of artifacts through carbon dating. These concepts are also used in computer science to analyze the efficiency of algorithms.
