Expressing Values As Powers Of 3: Step-by-Step Solutions

Hey guys! Today, we're diving into the fascinating world of exponents and how to express values as powers, specifically with a base of 3. This is a common type of problem in mathematics, and mastering it can really boost your problem-solving skills. We'll break down three different expressions step by step, so you can follow along and understand the logic behind each solution. Let's get started!

a) $3^{-2-0.6} \times 9^{2+0.3}$

Okay, let's tackle the first expression: $3^{-2-0.6} \times 9^{2+0.3}$. When we're dealing with exponents, the key is to get everything into the same base if possible. In this case, we want a base of 3. Notice that 9 is actually a power of 3; it's $3^2$. This is our starting point for simplifying the expression. When simplifying this expression, it’s crucial to remember the rules of exponents. Specifically, we'll use the rule that $(a^m)^n = a^{m \times n}$ when dealing with powers raised to other powers, and $a^m \times a^n = a^{m+n}$ when multiplying powers with the same base. First, let's simplify the exponents themselves. We have $-2 - 0.6$, which equals $-2.6$, and $2 + 0.3$, which equals $2.3$. So, our expression now looks like $3^{-2.6} \times 9^{2.3}$. Remember, we want to express everything in terms of base 3. We can rewrite $9^{2.3}$ as $(3^2)^{2.3}$. Using the power of a power rule, we multiply the exponents: $2 \times 2.3 = 4.6$. Now our expression is $3^{-2.6} \times 3^{4.6}$. Now that we have the same base, we can use the rule for multiplying exponents with the same base, which states that we add the exponents. So, we have $3^{-2.6 + 4.6}$, which simplifies to $3^2$. Thus, the expression simplifies to $3^2$, which is equal to 9. So, the value of the expression $3^{-2-0.6} \times 9^{2+0.3}$ expressed as a power of 3 is $3^2$. Understanding these steps thoroughly is important for tackling more complex exponent problems. Practice makes perfect, so try working through similar examples to solidify your understanding.

b) $(\frac{1}{3})^{-6} : 81^{0.2} : \frac{1}{3^8}$

Now, let's move on to the second expression: $(\frac{1}{3})^{-6} : 81^{0.2} : \frac{1}{3^8}$. Again, our goal is to express everything as a power of 3. We can start by rewriting the terms in the expression to have a base of 3. We know that $\frac{1}{3}$ can be written as $3^{-1}$, and 81 is $3^4$. So, let’s substitute these values into the expression. The expression becomes $(3^{-1})^{-6} : (3^4)^{0.2} : 3^{-8}$. Notice that we’ve also rewritten $\frac{1}{3^8}$ as $3^{-8}$. This makes it easier to work with the exponents. Remember, when dividing powers with the same base, we subtract the exponents. So, the colon symbol “:” here represents division. Now, let's simplify the exponents using the power of a power rule: $(3^{-1})^{-6}$ becomes $3^{(-1 \times -6)} = 3^6$, and $(3^4)^{0.2}$ becomes $3^{(4 \times 0.2)} = 3^{0.8}$. Our expression now looks like $3^6 : 3^{0.8} : 3^{-8}$. Next, we perform the divisions. Dividing $3^6$ by $3^{0.8}$ means subtracting the exponents: $6 - 0.8 = 5.2$. So, we have $3^{5.2} : 3^{-8}$. Now, we divide $3^{5.2}$ by $3^{-8}$, which means subtracting the exponents again: $5.2 - (-8) = 5.2 + 8 = 13.2$. So, the expression simplifies to $3^{13.2}$. Thus, the value of the expression $(\frac{1}{3})^{-6} : 81^{0.2} : \frac{1}{3^8}$ expressed as a power of 3 is $3^{13.2}$. This example highlights the importance of recognizing how to rewrite numbers as powers of a common base, which is a fundamental technique in simplifying exponential expressions.

c) $[(27^{-\frac{4}{9}})^{0.4}]^{-3}$

Finally, let's tackle the third expression: $[(27^{-\frac{4}{9}})^{0.4}]^{-3}$. This one looks a bit more complex, but don't worry, we'll break it down step by step. Our trusty method of expressing everything as a power of 3 will come in handy here as well. First, we need to recognize that 27 is $3^3$. So, we can rewrite the expression as $[((3^3)^{-\frac{4}{9}})^{0.4}]^{-3}$. This substitution is the key to simplifying the expression, as it allows us to apply the power of a power rule multiple times. Now, let's apply the power of a power rule. We have a series of exponents that we need to multiply together. Starting from the innermost exponent, we have $(3^3)^{-\frac{4}{9}}$, which simplifies to 3^{(3 \times -rac{4}{9})} = 3^{-\frac{4}{3}}. So, our expression now looks like $[(3^{-\frac{4}{3}})^{0.4}]^{-3}$. Next, we simplify $(3^{-\frac{4}{3}})^{0.4}$. Multiplying the exponents, we get -\frac{4}{3} \times 0.4 = -rac{4}{3} \times \frac{2}{5} = -rac{8}{15}. The expression becomes $[3^{-\frac{8}{15}}]^{-3}$. Finally, we apply the power of a power rule one more time: 3^{(-rac{8}{15} \times -3)} = 3^{\frac{24}{15}}. We can simplify the fraction $\frac{24}{15}$ by dividing both the numerator and denominator by 3, which gives us $\frac{8}{5}$. Thus, the expression simplifies to $3^{\frac{8}{5}}$. So, the value of the expression $[(27^{-\frac{4}{9}})^{0.4}]^{-3}$ expressed as a power of 3 is $3^{\frac{8}{5}}$. This example demonstrates how to handle nested exponents and fractions within exponents, which is a common challenge in algebra. By breaking the problem down into smaller steps and applying the power of a power rule consistently, we can simplify even the most complex expressions.

Conclusion

So, there you have it! We've successfully expressed the values of three different expressions as powers of 3. Remember, the key to solving these types of problems is to identify common bases and apply the rules of exponents correctly. Practice these steps, and you'll become a pro at simplifying exponential expressions! Keep up the great work, guys!