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The CENTRE for EDUCATION in MATHEMATICS and CON
Problem of the Week
Problem C
Exponential Expressions
two expressions:
Expression
A:72 ×7^(x)
Expression
B:441 ×2^(y)
Et
x and
y are positive integers, find all ordered pairs
(x,y) so that th Exprestion
A is equal to the value of Expression
B.

A=B

The CENTRE for EDUCATION in MATHEMATICS and CON $\newline$ Problem of the Week $\newline$ Problem C $\newline$ Exponential Expressions $\newline$ two expressions: $\newline$ Expression $A: 72 \times 7^{x}$ $\newline$ Expression $B: 441 \times 2^{y}$ $\newline$ Et $x$ and $y$ are positive integers, find all ordered pairs $(x, y)$ so that th Exprestion $A$ is equal to the value of Expression $B$ . $\newline$ $A=B$

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Write equations of cosine functions using properties

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Q. The CENTRE for EDUCATION in MATHEMATICS and CON

\newline

Problem of the Week

\newline

Problem C

\newline

Exponential Expressions

\newline

two expressions:

\newline

Expression

A: 72 \times 7^{x}

\newline

Expression

B: 441 \times 2^{y}

\newline

Et

x

and

y

are positive integers, find all ordered pairs

(x, y)

so that th Exprestion

A

is equal to the value of Expression

B

.

\newline

A=B

Factor out common factors: First, simplify the coefficients by factoring out common factors. $441$ is $63$ squared, and $63$ is $9$ times $7$ , so $441 = 9 \times 7 \times 7$ . Now we have $7^2 \times 7^{x} = 9 \times 7 \times 7 \times 2^{y}$ .
Divide by $9$ : Divide both sides by $9$ to simplify. $72 \div 9 = 8$ . So we get $8 \times 7^{x} = 7 \times 7 \times 2^{y}$ .
Isolate $7^{x}$ : Now, divide both sides by $7$ to isolate $7^{x}$ on one side. $8 \times 7^{x-1} = 7 \times 2^{y}.$
Rewrite $8$ as $2^3$ : We can see that $8$ is $2$ cubed, so we rewrite $8$ as $2^3$ . Now we have $2^3 \times 7^{(x-1)} = 7 \times 2^y$ .
Equate the exponents: Since the bases are now the same ( $2$ and $7$ ), we can equate the exponents. $3 + (x-1) = y$ and $x-1 = 1$ . So we have two equations: $3 + x - 1 = y$ and $x - 1 = 1$ .
Solve for x: Solve the second equation for x. $x - 1 = 1$ gives us $x = 2$ .
Substitute $x$ into $y$ : Substitute $x = 2$ into the first equation to find $y$ . $3 + 2 - 1 = y$ gives us $y = 4$ .
Check the solution: Check if the ordered pair $(x,y) = (2,4)$ satisfies the original equation. $72 \times 7^{2}$ should equal $441 \times 2^{4}$ . Calculate both sides: $72 \times 49 = 3528$ and $441 \times 16 = 7056$ . Oops, there's a mistake here, the calculations don't match.

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