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Magazine Chapter 1: Problem 28
Solve for \(x\). $$ 2^{5 x}=2^{x+8} $$
Short Answer
Expert verified
x = 2
Step by step solution
01
Recognize the Bases are Equal
Notice that both sides of the equation have the base of 2. This means we can apply the property of exponents which states that if the bases are equal, then the exponents must be equal if the equation is to hold true.
02
Set the Exponents Equal to Each Other
Since the bases are the same, set the exponents equal to each other: \[ 5x = x + 8 \].
03
Solve the Equation for x
Substract \(x\) from both sides of the equation to isolate x: \[ 5x - x = x + 8 - x \] which simplifies to: \[ 4x = 8 \].
04
Divide Both Sides by the Coefficient of x
Finally, divide both sides of the equation by 4 to solve for \(x\): \[ x = \frac{8}{4} \] which simplifies to: \[ x = 2 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponents
Exponents are a fundamental concept in algebra that represent repeated multiplication of a base number. When you see an expression like \( a^n \), it means that the base \( a \) is multiplied by itself \( n \) times. The exponent \( n \) tells you how many times the base is used as a factor. Understanding exponents is crucial because they appear frequently in various mathematical problems.
Here are some basic rules and operations associated with exponents:
Here are some basic rules and operations associated with exponents:
- Multiplying powers with the same base: Add the exponents, like \( a^m \times a^n = a^{m+n} \).
- Dividing powers with the same base: Subtract the exponents, such as \( \frac{a^m}{a^n} = a^{m-n} \).
- Power of a power: Multiply the exponents, e.g., \((a^m)^n = a^{m\cdot n} \).
- Zero exponent: Any base with an exponent of zero equals one, \( a^0 = 1 \).
Equation Solving
Solving equations is the process of finding the value of variables that make the equation true. In algebra, equations represent relationships between different quantities, and solving them is a fundamental skill.
When tackling equations like the one in the example \( 5x = x + 8 \), it's important to understand the steps involved:
When tackling equations like the one in the example \( 5x = x + 8 \), it's important to understand the steps involved:
- Identify like terms: Recognize and group similar variables together.
- Simplify the equation: Apply operations to both sides to isolate the variable.
- Use inverse operations: These are the opposite actions needed to simplify or solve (e.g., addition and subtraction are inverses).
- Check your solution: Substitute your answer back into the original equation to ensure it's correct.
Properties of Exponents
Understanding the properties of exponents is essential for solving problems that involve exponential expressions. These properties help simplify complex equations and make calculations more manageable. Let鈥檚 explore some key properties using exponential form:
- Product of Powers: When multiplying two exponential terms with the same base, keep the base and add the exponents: \( a^m \cdot a^n = a^{m+n} \).
- Quotient of Powers: When dividing two exponential terms with the same base, keep the base and subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
- Power of a Product: To raise a product to an exponent, raise each factor to the exponent separately: \((ab)^n = a^n \cdot b^n \).
- Power of a Quotient: To raise a quotient to an exponent, raise both the numerator and the denominator to the exponent: \((\frac{a}{b})^n = \frac{a^n}{b^n} \).
