In the expression below different letters stand for different one-digit numbers: $\newline$ $E \times V \times E \times R \times Y \times T \times H \times I \times N \times G \times I \times S \times G \times R \times E \times A \times T$ $\newline$ a) What is the maximum value the expression can have? $\newline$ b) What could be the maximum value, if we cross out one letter with all its repetitions?

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Algebra 2

Simplify radical expressions with variables II

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Q. In the expression below different letters stand for different one-digit numbers:

\newline

E \times V \times E \times R \times Y \times T \times H \times I \times N \times G \times I \times S \times G \times R \times E \times A \times T

\newline

a) What is the maximum value the expression can have?

\newline

b) What could be the maximum value, if we cross out one letter with all its repetitions?

Identify Letters: First, identify the letters in the expression: $E$ , $V$ , $R$ , $Y$ , $T$ , $H$ , $I$ , $N$ , $G$ , $S$ , $A$ .
Count Repetitions: Count the repetitions of each letter: E: $3$ , V: $1$ , R: $2$ , Y: $1$ , T: $2$ , H: $1$ , I: $2$ , N: $1$ , G: $2$ , S: $1$ , A: $1$ .
Assign Digits: To maximize the value, assign the highest digits ( $9$ , $8$ , $7$ , $6$ , $5$ , $4$ , $3$ , $2$ , $1$ ) to the most frequent letters. E: $9$ , R: $8$ , T: $7$ , I: $6$ , G: $5$ , V: $4$ , Y: $3$ , H: $2$ , N: $1$ , S: $0$ , A: $0$ .
Calculate Product: Calculate the product: $9 \times 4 \times 9 \times 8 \times 3 \times 7 \times 2 \times 6 \times 1 \times 5 \times 6 \times 0 \times 5 \times 8 \times 9 \times 0 \times 7$ .
Check for Zeros: Since there are zeros in the product, the maximum value is $0$ .
Cross Out Letter: For part b, cross out one letter with all its repetitions to avoid zeros. Cross out S $(0)$ or A $(0)$ .
Recalculate Without S: Recalculate without S: $9 \times 4 \times 9 \times 8 \times 3 \times 7 \times 2 \times 6 \times 1 \times 5 \times 6 \times 5 \times 8 \times 9 \times 7$ .
Recalculate Without A: Recalculate without A: $9 \times 4 \times 9 \times 8 \times 3 \times 7 \times 2 \times 6 \times 1 \times 5 \times 6 \times 5 \times 8 \times 9 \times 7$ .
Same Product: Both give the same product: $= 9 \times 4 \times 9 \times 8 \times 3 \times 7 \times 2 \times 6 \times 1 \times 5 \times 6 \times 5 \times 8 \times 9 \times 7$ .
Calculate Step by Step: Calculate step by step:
$9 \times 4 = 36$
$36 \times 9 = 324$
$324 \times 8 = 2592$
$2592 \times 3 = 7776$
$7776 \times 7 = 54432$
$54432 \times 2 = 108864$
$108864 \times 6 = 653184$
$653184 \times 1 = 653184$
$653184 \times 5 = 3265920$
$3265920 \times 6 = 19595520$
$19595520 \times 5 = 97977600$
$97977600 \times 8 = 783820800$
$783820800 \times 9 = 7054387200$
$7054387200 \times 7 = 49380710400$ .