Algebra Problems: Exponents, Equations, And Inequalities

Hey guys! Let's dive into some cool algebra problems focusing on exponents, equations, and inequalities. We'll break down each problem step-by-step so you can totally nail it. So, grab your pencils, and let's get started!

1. Calculating Expressions with Exponents

In this section, we will focus on calculating expressions with exponents, including negative exponents, fractional exponents, and combinations of these. Understanding exponents is super crucial in algebra, and it's something you'll use a lot, so let’s make sure we get it down. We will explore different types of exponents and how they affect the outcome of our calculations. Let's tackle each part one by one. First, we'll discuss negative exponents, which might seem a bit tricky at first, but they're actually quite straightforward once you understand the rule. Then we'll move on to fractional exponents, which are closely related to radicals and roots. Finally, we'll combine these concepts to solve more complex expressions. By the end of this section, you'll be an exponent expert, ready to tackle any problem that comes your way. Remember, math is all about practice, so don't be afraid to try out different approaches and learn from your mistakes. Let's jump in and start calculating!

a) Calculate $2^{-6}$

When you see a negative exponent, like in the expression $2^{-6}$, it means you're dealing with the reciprocal of the base raised to the positive exponent. In simpler terms, $a^{-n} = \frac{1}{a^n}$. So, for $2^{-6}$, it means we need to find the reciprocal of $2^6$. Let's break it down:

• First, calculate $2^6$, which is $2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$.
• Then, take the reciprocal: $\frac{1}{64}$.

So, $2^{-6} = \frac{1}{64}$. See? Negative exponents aren't so scary after all! It's all about flipping the base and changing the sign of the exponent. Keep this rule in mind, and you'll be solving negative exponent problems like a pro. This concept is super useful in various areas of math, so mastering it now will definitely pay off later. Now, let's move on to the next part of the problem and tackle fractional exponents. Get ready, because we're about to level up our exponent skills!

b) Calculate $(\frac{3}{7})^{-1}$

Now, let's tackle another negative exponent, but this time, it's applied to a fraction: $(\frac{3}{7})^{-1}$. Remember the rule we just learned? A negative exponent means we take the reciprocal. When you have a fraction raised to the power of -1, you simply flip the fraction. So, $(\frac{a}{b})^{-1} = \frac{b}{a}$. Applying this to our problem:

• $(\frac{3}{7})^{-1}$ simply becomes $\frac{7}{3}$.

That's it! Super straightforward, right? The negative exponent just tells us to flip the fraction. This is a handy shortcut to remember when you're dealing with fractions and negative exponents. It saves you the step of calculating the reciprocal separately. Now, let's move on to something a bit more challenging: expressions with both fractional and integer exponents. We're building up our exponent toolkit, one step at a time. Keep practicing, and you'll be amazed at how quickly you can solve these problems. Next up, we have a combination of fractional exponents and subtraction. Let's see how we can handle that!

c) Calculate $125^{\frac{1}{3}}-64^{\frac{1}{6}}$

Okay, this one looks a bit more interesting! We have fractional exponents and subtraction in the mix: $125^{\frac{1}{3}}-64^{\frac{1}{6}}$. Fractional exponents are closely related to roots. Specifically, $a^{\frac{1}{n}}$ is the same as the nth root of a, which is written as $\sqrt[n]{a}$. So, let's rewrite our expression using roots:

• $125^{\frac{1}{3}}$ is the same as $\sqrt[3]{125}$, which is the cube root of 125.
• $64^{\frac{1}{6}}$ is the same as $\sqrt[6]{64}$, which is the sixth root of 64.

Now, let's calculate these roots:

• The cube root of 125 is 5 because $5 \times 5 \times 5 = 125$.
• The sixth root of 64 is 2 because $2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$.

Now, substitute these values back into our expression:

• $125^{\frac{1}{3}}-64^{\frac{1}{6}} = 5 - 2 = 3$.

So, the final answer is 3. Nice job! We tackled fractional exponents by converting them to roots, which made the calculation much easier. Remember, when you see a fractional exponent, think roots! This will help you simplify the expression and find the answer. We've covered negative exponents, reciprocals, and fractional exponents. You're becoming quite the exponent master! Now, let's move on to solving equations with exponents.

2. Solving Exponential Equations

Moving on, let's talk about solving exponential equations. These are equations where the variable appears in the exponent. The main goal here is to get both sides of the equation to have the same base, so you can then equate the exponents. This often involves some clever manipulation and knowing your exponent rules inside and out. It’s like detective work, but with numbers! There are several techniques we can use, such as rewriting numbers as powers of a common base, using logarithms, or applying exponent rules to simplify the equation. Each equation might require a slightly different approach, so it's important to have a good understanding of the underlying principles. We’ll go through several examples to illustrate these techniques and show you how to tackle different types of exponential equations. So, buckle up, and let’s solve some equations! Remember, the key to success is practice, so don't hesitate to try out different methods and see what works best for you. Let's start with our first equation and break it down step by step.

a) Solve $8^x = 4096$

To solve the equation $8^x = 4096$, we need to express both sides with the same base. Notice that both 8 and 4096 can be written as powers of 2. Let's do that:

• $8 = 2^3$, so we can rewrite $8^x$ as $(2^3)^x = 2^{3x}$.
• Now, we need to find what power of 2 equals 4096. If you calculate it, you'll find that $4096 = 2^{12}$.

So, our equation becomes $2^{3x} = 2^{12}$. Now that we have the same base on both sides, we can equate the exponents:

• $3x = 12$
• Divide both sides by 3 to solve for x: $x = \frac{12}{3} = 4$.

So, the solution to the equation is $x = 4$. Great job! By expressing both sides with the same base, we turned a tricky exponential equation into a simple linear equation. This is a common strategy when solving these types of problems. Keep an eye out for common bases, and you'll be able to solve these equations with ease. Now, let's move on to the next equation, which involves a slightly different twist. Get ready to use your exponent-solving skills again!

b) Solve $19^{x^2-10x} = 1$

Here we have the equation $19^{x^2-10x} = 1$. To solve this, remember that any non-zero number raised to the power of 0 is 1. So, we can rewrite the right side of the equation as $19^0$. This gives us:

• $19^{x^2-10x} = 19^0$

Now that we have the same base, we can equate the exponents:

• $x^2 - 10x = 0$

This is a quadratic equation. To solve it, we can factor out an x:

• $x(x - 10) = 0$

This gives us two possible solutions:

• $x = 0$ or $x - 10 = 0$, which means $x = 10$.

So, the solutions to the equation are $x = 0$ and $x = 10$. Excellent! We used the property of exponents that anything to the power of 0 is 1 to simplify the equation and then solved the resulting quadratic equation. This problem highlights the importance of knowing your exponent properties and how they can help you solve equations. Let's move on to the next equation, which involves a decimal base and a bit more manipulation. Keep up the great work!

c) Solve $(0,2)^{2-3x} = 25$

This equation, $(0,2)^{2-3x} = 25$, looks a bit trickier, but we can totally handle it. First, let's rewrite 0.2 as a fraction: $0.2 = \frac{1}{5}$. So, our equation becomes:

• $(\frac{1}{5})^{2-3x} = 25$

Now, let's rewrite both sides with the same base. Notice that 25 is $5^2$, and $\frac{1}{5}$ is $5^{-1}$. So, we can rewrite the equation as:

• $(5^{-1})^{2-3x} = 5^2$

Using the exponent rule $(a^m)^n = a^{mn}$, we get:

• $5^{-(2-3x)} = 5^2$

Now, we can equate the exponents:

• $-(2 - 3x) = 2$
• $-2 + 3x = 2$
• $3x = 4$
• $x = \frac{4}{3}$

So, the solution to the equation is $x = \frac{4}{3}$. Fantastic! We tackled this equation by rewriting the decimal as a fraction, expressing both sides with the same base, and then equating the exponents. This problem demonstrates how flexible you need to be when solving exponential equations, using different strategies to simplify the problem. Let's move on to our final equation in this section, which involves a little bit of factoring. You're doing great!

d) Solve $4^{x+3}-4^x = 63$

Okay, let's dive into the last equation: $4^{x+3}-4^x = 63$. To solve this, we'll use a bit of factoring. Notice that $4^{x+3}$ can be rewritten as $4^x \times 4^3$ using the exponent rule $a^{m+n} = a^m \times a^n$. So, our equation becomes:

• $4^x \times 4^3 - 4^x = 63$

Now, we can factor out $4^x$ from both terms:

• $4^x(4^3 - 1) = 63$
• $4^x(64 - 1) = 63$
• $4^x(63) = 63$

Divide both sides by 63:

• $4^x = 1$

Remember that any non-zero number raised to the power of 0 is 1, so:

• $x = 0$

So, the solution to the equation is $x = 0$. Awesome! We solved this equation by using factoring and the property that anything to the power of 0 is 1. This problem shows how important it is to look for common factors when solving equations. You've now tackled a variety of exponential equations, using different techniques and strategies. You're well on your way to becoming an exponential equation-solving master! Now, let's switch gears and move on to solving inequalities with exponents.

3. Solving Exponential Inequalities

Now, let's jump into solving exponential inequalities. Inequalities are like equations, but instead of an equals sign, we have signs like >, <, ≥, or ≤. When dealing with exponential inequalities, the key is to again try to get the same base on both sides. However, there's a little twist: you need to consider whether the base is greater than 1 or between 0 and 1. If the base is greater than 1, the inequality sign stays the same when you equate the exponents. But, if the base is between 0 and 1, the inequality sign flips! This is a super important detail to remember. We'll walk through an example to make this clear. We'll also look at how to manipulate the inequality to get it into a form where we can easily compare the exponents. Inequalities are a fundamental concept in algebra, and understanding how to solve exponential inequalities will give you a powerful tool for problem-solving. So, let’s dive in and see how it’s done! Get ready to sharpen your inequality skills. Let's tackle the inequality step by step and make sure we understand the rules.

a) Solve $7^x > 343$

To solve the inequality $7^x > 343$, we need to express both sides with the same base, just like we did with equations. We know that 343 is a power of 7. Specifically, $343 = 7^3$. So, we can rewrite the inequality as:

• $7^x > 7^3$

Since the base (7) is greater than 1, the inequality sign stays the same when we equate the exponents. This means:

• $x > 3$

So, the solution to the inequality is $x > 3$. This means any value of x greater than 3 will satisfy the inequality. That was pretty straightforward, right? When the base is greater than 1, solving exponential inequalities is very similar to solving equations. You just need to make sure you keep the inequality sign the same. However, remember the rule about the base being between 0 and 1! That's a crucial detail. You’ve successfully solved an exponential inequality! Now, you have a solid understanding of how to tackle these types of problems. Remember, practice makes perfect, so keep working on different examples to build your skills. And that's a wrap, guys! You've done an amazing job working through these algebra problems. Keep practicing, and you'll become a math whiz in no time!