1. Between which two integers do the following surds lie? $$ \begin{array}{ll} ext { a) } \sqrt{56} & ext { b) }-\sqrt{12} \ ext { c) }-\sqrt{78} & ext { d) } \sqrt{15} \ ext { e) }-\sqrt{43} & ext { f) }-\sqrt{29} \ ext { g) } \sqrt{99} & ext { h) } \sqrt{8}\end{array} $$

Question

1. Between which two integers do the following surds lie? $$ \begin{array}{ll}	ext { a) } \sqrt{56} & 	ext { b) }-\sqrt{12} \ 	ext { c) }-\sqrt{78} & 	ext { d) } \sqrt{15} \ 	ext { e) }-\sqrt{43} & 	ext { f) }-\sqrt{29} \ 	ext { g) } \sqrt{99} & 	ext { h) } \sqrt{8}\end{array} $$

UpStudy AI · Accepted Answer

Let's determine between which two consecutive integers each surd lies by evaluating their approximate values.

### a) $$ \sqrt{56} $$
- **Calculation**: $$7^2 = 49$$ and $$8^2 = 64$$.
- **Conclusion**: $$7 < \sqrt{56} < 8$$.

### b) $$ -\sqrt{12} $$
- **Calculation**: $$ \sqrt{12} \approx 3.464 $$.
- **Conclusion**: $$-4 < -\sqrt{12} < -3$$.

### c) $$ -\sqrt{78} $$
- **Calculation**: $$ \sqrt{78} \approx 8.832 $$.
- **Conclusion**: $$-9 < -\sqrt{78} < -8$$.

### d) $$ \sqrt{15} $$
- **Calculation**: $$3^2 = 9$$ and $$4^2 = 16$$.
- **Conclusion**: $$3 < \sqrt{15} < 4$$.

### e) $$ -\sqrt{43} $$
- **Calculation**: $$ \sqrt{43} \approx 6.557 $$.
- **Conclusion**: $$-7 < -\sqrt{43} < -6$$.

### f) $$ -\sqrt{29} $$
- **Calculation**: $$ \sqrt{29} \approx 5.385 $$.
- **Conclusion**: $$-6 < -\sqrt{29} < -5$$.

### g) $$ \sqrt{99} $$
- **Calculation**: $$9^2 = 81$$ and $$10^2 = 100$$.
- **Conclusion**: $$9 < \sqrt{99} < 10$$.

### h) $$ \sqrt{8} $$
- **Calculation**: $$2^2 = 4$$ and $$3^2 = 9$$.
- **Conclusion**: $$2 < \sqrt{8} < 3$$.

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**Summary of Results:**

a) $$7 < \sqrt{56} < 8$$  
b) $$-4 < -\sqrt{12} < -3$$  
c) $$-9 < -\sqrt{78} < -8$$  
d) $$3 < \sqrt{15} < 4$$  
e) $$-7 < -\sqrt{43} < -6$$  
f) $$-6 < -\sqrt{29} < -5$$  
g) $$9 < \sqrt{99} < 10$$  
h) $$2 < \sqrt{8} < 3$$
Here are the ranges for each surd: a) $$7 < \sqrt{56} < 8$$ b) $$-4 < -\sqrt{12} < -3$$ c) $$-9 < -\sqrt{78} < -8$$ d) $$3 < \sqrt{15} < 4$$ e) $$-7 < -\sqrt{43} < -6$$ f) $$-6 < -\sqrt{29} < -5$$ g) $$9 < \sqrt{99} < 10$$ h) $$2 < \sqrt{8} < 3$$

Between which two integers do the following surds lie?

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