Evaluate

6log2^3 √64+ 10log3^5 √243

6log2^3 √64+ 10log3^5 √243

= 6log(8*8) + 10log(243√243)
= log 64^10 + log ( (243√243)^10)
= log ( (64^10)(243√243)^10 )
= log ( (64^10)(243^15) )
= log (7.0128 x 10^35)
= appr 53.85

I suspect a typo

yes there is a typo

6 log2 ∛64 + 10log3 (243)^1/5

To evaluate the given expression step-by-step, we will start by simplifying the logarithmic expressions:

1. Simplify log2^3 √64:
- Since 64 is equal to 2^6, then √64 = √(2^6) = 2^3 = 8.
- Therefore, log2^3 √64 = log2^3 8.

2. Evaluate log2^3 8:
- To simplify this logarithmic expression, we need to rewrite it in exponential form.
- Since 2^3 = 8, then log2^3 8 = 3.

3. Simplify log3^5 √243:
- Since 243 is equal to 3^5, then √243 = √(3^5) = 3^(5/2).
- Therefore, log3^5 √243 = log3^5 3^(5/2).

4. Evaluate log3^5 3^(5/2):
- To simplify this logarithmic expression, we need to rewrite it in exponential form.
- Since 3^(5/2) = (3^5)^(1/2), then log3^5 3^(5/2) = 5 * (1/2) = 5/2.

5. Finally, substitute the simplified values into the given expression:
6log2^3 √64 + 10log3^5 √243
= 6 * 3 + 10 * (5/2)
= 18 + 25
= 43.

Therefore, the value of the given expression is 43.

To evaluate the expression, let's break it down step by step.

Step 1: Simplify the logarithmic expressions.
- Recall that log rules state that log_a(b^c) = c * log_a(b). Using this rule, we can simplify the logarithmic terms in the expression.

6log2^3 = 6 * 3 * log2 = 18log2
10log3^5 = 10 * 5 * log3 = 50log3

Step 2: Simplify the square root terms.
- We can simplify the square roots by finding the square roots of 64 and 243.

√64 = 8
√243 = 9√3

Step 3: Substitute the simplified terms back into the expression.
Now, we can substitute the simplified logarithmic and square root terms back into the original expression.

18log2√64 + 50log3√243 = 18log2(8) + 50log3(9√3)

Step 4: Evaluate the logarithmic and square root terms.
- Using the properties of logarithms and simplifying the square roots, we can further evaluate the expression.

18log2(8) = 18log2(2^3) = 18 * 3 = 54
50log3(9√3) = 50log3(9) + 50log3(√3) = 50 * 2 + 50 * (1/2)log3(3) = 100 + 25log3(3) = 100 + 25 * 1 = 125

Step 5: Add the results together.
Finally, add the results from step 4.

54 + 125 = 179

Therefore, the evaluated expression is 179.