Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Certainly! Let's go through each expression step-by-step to rewrite them in terms of [tex]\( y = 3^x \)[/tex].
### (a) [tex]\( 3^{3x} \)[/tex]
To express [tex]\( 3^{3x} \)[/tex] in terms of [tex]\( y \)[/tex], we use the properties of exponents. Specifically, we know that:
[tex]\[ 3^{3x} = (3^x)^3 \][/tex]
Since [tex]\( y = 3^x \)[/tex], we substitute [tex]\( y \)[/tex] for [tex]\( 3^x \)[/tex]:
[tex]\[ 3^{3x} = (3^x)^3 = y^3 \][/tex]
Thus:
[tex]\[ 3^{3x} = y^3 \][/tex]
### (b) [tex]\( \frac{1}{3^{x-2}} \)[/tex]
To express [tex]\( \frac{1}{3^{x-2}} \)[/tex] in terms of [tex]\( y \)[/tex], we can simplify this using properties of exponents. First, recall that [tex]\(\frac{1}{a^b} = a^{-b}\)[/tex]:
[tex]\[ \frac{1}{3^{x-2}} = 3^{-(x-2)} \][/tex]
Next, simplify the exponent:
[tex]\[ 3^{-(x-2)} = 3^{-x+2} \][/tex]
This can be rewritten as:
[tex]\[ 3^{-x+2} = 3^{-x} \cdot 3^2 \][/tex]
Using [tex]\( y = 3^x \)[/tex] implies [tex]\( 3^{-x} = \frac{1}{y} \)[/tex]:
[tex]\[ 3^{-x+2} = \frac{1}{y} \cdot 9 = \frac{9}{y} \][/tex]
Thus:
[tex]\[ \frac{1}{3^{x-2}} = \frac{9}{y} \][/tex]
### (c) [tex]\( \frac{81}{9^{2-3x}} \)[/tex]
To express [tex]\( \frac{81}{9^{2-3x}} \)[/tex] in terms of [tex]\( y \)[/tex], we will start by rewriting 81 and 9 in terms of base 3:
[tex]\[ 81 = 3^4 \][/tex]
[tex]\[ 9 = 3^2 \][/tex]
Thus:
[tex]\[ 9^{2-3x} = (3^2)^{2-3x} = 3^{2(2-3x)} = 3^{4-6x} \][/tex]
Now, rewrite the entire expression:
[tex]\[ \frac{81}{9^{2-3x}} = \frac{3^4}{3^{4-6x}} \][/tex]
Using the properties of exponents, specifically [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], we get:
[tex]\[ \frac{3^4}{3^{4-6x}} = 3^{4-(4-6x)} = 3^{4-4+6x} = 3^{6x} \][/tex]
Since [tex]\( y = 3^x \)[/tex], we have:
[tex]\[ 3^{6x} = (3^x)^6 = y^6 \][/tex]
Thus:
[tex]\[ \frac{81}{9^{2-3x}} = y^6 \][/tex]
In conclusion, the expressions in their simplest forms in terms of [tex]\( y \)[/tex] are:
(a) [tex]\( 3^{3x} = y^3 \)[/tex]
(b) [tex]\( \frac{1}{3^{x-2}} = \frac{9}{y} \)[/tex]
(c) [tex]\( \frac{81}{9^{2-3x}} = y^6 \)[/tex]
### (a) [tex]\( 3^{3x} \)[/tex]
To express [tex]\( 3^{3x} \)[/tex] in terms of [tex]\( y \)[/tex], we use the properties of exponents. Specifically, we know that:
[tex]\[ 3^{3x} = (3^x)^3 \][/tex]
Since [tex]\( y = 3^x \)[/tex], we substitute [tex]\( y \)[/tex] for [tex]\( 3^x \)[/tex]:
[tex]\[ 3^{3x} = (3^x)^3 = y^3 \][/tex]
Thus:
[tex]\[ 3^{3x} = y^3 \][/tex]
### (b) [tex]\( \frac{1}{3^{x-2}} \)[/tex]
To express [tex]\( \frac{1}{3^{x-2}} \)[/tex] in terms of [tex]\( y \)[/tex], we can simplify this using properties of exponents. First, recall that [tex]\(\frac{1}{a^b} = a^{-b}\)[/tex]:
[tex]\[ \frac{1}{3^{x-2}} = 3^{-(x-2)} \][/tex]
Next, simplify the exponent:
[tex]\[ 3^{-(x-2)} = 3^{-x+2} \][/tex]
This can be rewritten as:
[tex]\[ 3^{-x+2} = 3^{-x} \cdot 3^2 \][/tex]
Using [tex]\( y = 3^x \)[/tex] implies [tex]\( 3^{-x} = \frac{1}{y} \)[/tex]:
[tex]\[ 3^{-x+2} = \frac{1}{y} \cdot 9 = \frac{9}{y} \][/tex]
Thus:
[tex]\[ \frac{1}{3^{x-2}} = \frac{9}{y} \][/tex]
### (c) [tex]\( \frac{81}{9^{2-3x}} \)[/tex]
To express [tex]\( \frac{81}{9^{2-3x}} \)[/tex] in terms of [tex]\( y \)[/tex], we will start by rewriting 81 and 9 in terms of base 3:
[tex]\[ 81 = 3^4 \][/tex]
[tex]\[ 9 = 3^2 \][/tex]
Thus:
[tex]\[ 9^{2-3x} = (3^2)^{2-3x} = 3^{2(2-3x)} = 3^{4-6x} \][/tex]
Now, rewrite the entire expression:
[tex]\[ \frac{81}{9^{2-3x}} = \frac{3^4}{3^{4-6x}} \][/tex]
Using the properties of exponents, specifically [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], we get:
[tex]\[ \frac{3^4}{3^{4-6x}} = 3^{4-(4-6x)} = 3^{4-4+6x} = 3^{6x} \][/tex]
Since [tex]\( y = 3^x \)[/tex], we have:
[tex]\[ 3^{6x} = (3^x)^6 = y^6 \][/tex]
Thus:
[tex]\[ \frac{81}{9^{2-3x}} = y^6 \][/tex]
In conclusion, the expressions in their simplest forms in terms of [tex]\( y \)[/tex] are:
(a) [tex]\( 3^{3x} = y^3 \)[/tex]
(b) [tex]\( \frac{1}{3^{x-2}} = \frac{9}{y} \)[/tex]
(c) [tex]\( \frac{81}{9^{2-3x}} = y^6 \)[/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
